appease, make (an atonement, cleanse, disannul, forgive, be merciful, pacify, pardon, purge (away), put off, (make) reconcile(-liation)

Boyd’s Bible Dictionary:

“Forgiveness” From Concise Bible Dictionary:

There are three Hebrew words translated to forgive.
1. kaphar, “to cover” (Deut. 21:88Be merciful, O Lord, unto thy people Israel, whom thou hast redeemed, and lay not innocent blood unto thy people of Israel's charge. And the blood shall be forgiven them. (Deuteronomy 21:8); Psa. 78:3838But he, being full of compassion, forgave their iniquity, and destroyed them not: yea, many a time turned he his anger away, and did not stir up all his wrath. (Psalm 78:38); Jer. 18:2323Yet, Lord, thou knowest all their counsel against me to slay me: forgive not their iniquity, neither blot out their sin from thy sight, but let them be overthrown before thee; deal thus with them in the time of thine anger. (Jeremiah 18:23)). It is also translated “atonement.”
2. nasa, “to bear,” take away [guilt]: used by Joseph's brethren when they asked him to forgive them (Gen. 50:1717So shall ye say unto Joseph, Forgive, I pray thee now, the trespass of thy brethren, and their sin; for they did unto thee evil: and now, we pray thee, forgive the trespass of the servants of the God of thy father. And Joseph wept when they spake unto him. (Genesis 50:17)); and used of God as "forgiving iniquity and transgression and sin" (Ex. 34:77Keeping mercy for thousands, forgiving iniquity and transgression and sin, and that will by no means clear the guilty; visiting the iniquity of the fathers upon the children, and upon the children's children, unto the third and to the fourth generation. (Exodus 34:7); Num. 14:1818The Lord is longsuffering, and of great mercy, forgiving iniquity and transgression, and by no means clearing the guilty, visiting the iniquity of the fathers upon the children unto the third and fourth generation. (Numbers 14:18)); and in describing the blessedness of the man "whose transgression is forgiven, whose sin is covered" (Psa. 32:11<<A Psalm of David, Maschil.>> Blessed is he whose transgression is forgiven, whose sin is covered. (Psalm 32:1)).
3. salach, “to pardon,” used only of the forgiveness that God gives. It is employed for the forgiveness attached to the sacrifices: "it shall be forgiven him" (Lev. 4:20, 26, 31, 3520And he shall do with the bullock as he did with the bullock for a sin offering, so shall he do with this: and the priest shall make an atonement for them, and it shall be forgiven them. (Leviticus 4:20)
26And he shall burn all his fat upon the altar, as the fat of the sacrifice of peace offerings: and the priest shall make an atonement for him as concerning his sin, and it shall be forgiven him. (Leviticus 4:26)
31And he shall take away all the fat thereof, as the fat is taken away from off the sacrifice of peace offerings; and the priest shall burn it upon the altar for a sweet savor unto the Lord; and the priest shall make an atonement for him, and it shall be forgiven him. (Leviticus 4:31)
35And he shall take away all the fat thereof, as the fat of the lamb is taken away from the sacrifice of the peace offerings; and the priest shall burn them upon the altar, according to the offerings made by fire unto the Lord: and the priest shall make an atonement for his sin that he hath committed, and it shall be forgiven him. (Leviticus 4:35); Lev. 5:10, 13, 16, 1810And he shall offer the second for a burnt offering, according to the manner: and the priest shall make an atonement for him for his sin which he hath sinned, and it shall be forgiven him. (Leviticus 5:10)
13And the priest shall make an atonement for him as touching his sin that he hath sinned in one of these, and it shall be forgiven him: and the remnant shall be the priest's, as a meat offering. (Leviticus 5:13)
16And he shall make amends for the harm that he hath done in the holy thing, and shall add the fifth part thereto, and give it unto the priest: and the priest shall make an atonement for him with the ram of the trespass offering, and it shall be forgiven him. (Leviticus 5:16)
18And he shall bring a ram without blemish out of the flock, with thy estimation, for a trespass offering, unto the priest: and the priest shall make an atonement for him concerning his ignorance wherein he erred and wist it not, and it shall be forgiven him. (Leviticus 5:18); &c). It occurs in the prayer of Solomon at the dedication of the temple (1 Kings 8:30, 34, 36, 39, 5030And hearken thou to the supplication of thy servant, and of thy people Israel, when they shall pray toward this place: and hear thou in heaven thy dwelling place: and when thou hearest, forgive. (1 Kings 8:30)
34Then hear thou in heaven, and forgive the sin of thy people Israel, and bring them again unto the land which thou gavest unto their fathers. (1 Kings 8:34)
36Then hear thou in heaven, and forgive the sin of thy servants, and of thy people Israel, that thou teach them the good way wherein they should walk, and give rain upon thy land, which thou hast given to thy people for an inheritance. (1 Kings 8:36)
39Then hear thou in heaven thy dwelling place, and forgive, and do, and give to every man according to his ways, whose heart thou knowest; (for thou, even thou only, knowest the hearts of all the children of men;) (1 Kings 8:39)
50And forgive thy people that have sinned against thee, and all their transgressions wherein they have transgressed against thee, and give them compassion before them who carried them captive, that they may have compassion on them: (1 Kings 8:50). Also in Psa. 103:33Who forgiveth all thine iniquities; who healeth all thy diseases; (Psalm 103:3); Jer. 31:3434And they shall teach no more every man his neighbor, and every man his brother, saying, Know the Lord: for they shall all know me, from the least of them unto the greatest of them, saith the Lord: for I will forgive their iniquity, and I will remember their sin no more. (Jeremiah 31:34); Jer. 36:33It may be that the house of Judah will hear all the evil which I purpose to do unto them; that they may return every man from his evil way; that I may forgive their iniquity and their sin. (Jeremiah 36:3); Dan. 9:1919O Lord, hear; O Lord, forgive; O Lord, hearken and do; defer not, for thine own sake, O my God: for thy city and thy people are called by thy name. (Daniel 9:19)).
In the N. T. two words are used: ὔφεσις., from ἀφίημι, “to send from, release, remit,” several times translated REMISSION; and χαριξομαι, “to be gracious, bestow freely, forgive.” Both words are applied to the forgiveness granted by God, as well as that between man and his fellow.
There are two aspects in which forgiveness is brought before us in scripture.
1. The mind and thought of God Himself towards the sinner whom He forgives. On the ground of the sacrifice of Christ, God not only ceases to hold those who have faith in Christ's blood as guilty before Him, but His favor is towards them. "Their sins and iniquities will I remember no more" (Heb. 10:1717And their sins and iniquities will I remember no more. (Hebrews 10:17)). Thus all sense of imputation of guilt is gone from the mind of God. "God for Christ's sake hath forgiven you" ἐχαρίσυτο, graciously forgiven, (Eph. 4:3232And be ye kind one to another, tenderhearted, forgiving one another, even as God for Christ's sake hath forgiven you. (Ephesians 4:32)). So in the O.T., "I will heal their backsliding, I will love them freely" (Hos. 14:44I will heal their backsliding, I will love them freely: for mine anger is turned away from him. (Hosea 14:4)).
2. The guilty one is released, forgiven. "That they may receive forgiveness of sins" (Acts 26:1818To open their eyes, and to turn them from darkness to light, and from the power of Satan unto God, that they may receive forgiveness of sins, and inheritance among them which are sanctified by faith that is in me. (Acts 26:18)). "As far as the east is from the west, so far hath he removed our transgressions from us" (Psa. 103:1212As far as the east is from the west, so far hath he removed our transgressions from us. (Psalm 103:12)). "Your sins are forgiven you for his name's sake" (1 John 2:1212I write unto you, little children, because your sins are forgiven you for his name's sake. (1 John 2:12)). Hence it is true of all Christians, that their sins are forgiven. Another thought is included in the forgiveness of sins, namely, that having redemption by Christ, which brings into a new state, the whole guilty past is forgiven, removed from us, so that there is no hindrance to the enjoyment of that into which redemption brings.
The general principle as to forgiveness is stated in 1 John 1:99If we confess our sins, he is faithful and just to forgive us our sins, and to cleanse us from all unrighteousness. (1 John 1:9): "If we confess our sins, he is faithful and just to forgive us our sins;" and to this is added, "and to cleanse us from all unrighteousness." This involves honesty of heart, whether in a sinner first coming to God, or in a child who has grieved the heart of the Father by sinning. The two aspects above referred to are here also. The faithfulness and righteousness of God in forgiving, and the cleansing us from all unrighteousness. God is faithful to His own blessed character of grace revealed in His Son, and righteous through the propitiation which He has made.
3. If a Christian is “put away” from the assembly and is repentant, he is forgiven and restored (2 Cor. 2:7, 107So that contrariwise ye ought rather to forgive him, and comfort him, lest perhaps such a one should be swallowed up with overmuch sorrow. (2 Corinthians 2:7)
10To whom ye forgive any thing, I forgive also: for if I forgave any thing, to whom I forgave it, for your sakes forgave I it in the person of Christ; (2 Corinthians 2:10)). This of course is different from the act of God in forgiving sins, and may be called administrative forgiveness in the church; and if the act of discipline is led of the Spirit, it is ratified in heaven: (Compare John 20:22-2322And when he had said this, he breathed on them, and saith unto them, Receive ye the Holy Ghost: 23Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained. (John 20:22‑23)). This is entirely different from any pretended absolution that may be pronounced over poor deluded unconverted persons.
4. There is also a governmental forgiveness in connection with the government of God here below in time, both on God's part, and toward one another (Isa. 40:1-21Comfort ye, comfort ye my people, saith your God. 2Speak ye comfortably to Jerusalem, and cry unto her, that her warfare is accomplished, that her iniquity is pardoned: for she hath received of the Lord's hand double for all her sins. (Isaiah 40:1‑2); Luke 17:33Take heed to yourselves: If thy brother trespass against thee, rebuke him; and if he repent, forgive him. (Luke 17:3); James 5:15-1615And the prayer of faith shall save the sick, and the Lord shall raise him up; and if he have committed sins, they shall be forgiven him. 16Confess your faults one to another, and pray one for another, that ye may be healed. The effectual fervent prayer of a righteous man availeth much. (James 5:15‑16); 1 John 5:1616If any man see his brother sin a sin which is not unto death, he shall ask, and he shall give him life for them that sin not unto death. There is a sin unto death: I do not say that he shall pray for it. (1 John 5:16)). We are called upon to forgive one another; and if we indulge in a harsh unforgiving spirit, we must not expect our Father to forgive us in His governmental dealings (Matt. 6:14-1514For if ye forgive men their trespasses, your heavenly Father will also forgive you: 15But if ye forgive not men their trespasses, neither will your Father forgive your trespasses. (Matthew 6:14‑15)).

“Reconciliation” From Concise Bible Dictionary:

Except in 1 Samuel 29:44And the princes of the Philistines were wroth with him; and the princes of the Philistines said unto him, Make this fellow return, that he may go again to his place which thou hast appointed him, and let him not go down with us to battle, lest in the battle he be an adversary to us: for wherewith should he reconcile himself unto his master? should it not be with the heads of these men? (1 Samuel 29:4), and 2 Chronicles 29:2424And the priests killed them, and they made reconciliation with their blood upon the altar, to make an atonement for all Israel: for the king commanded that the burnt offering and the sin offering should be made for all Israel. (2 Chronicles 29:24), the Hebrew word is kaphar, which is more than sixty times translated “to make an atonement;” and this rendering suits sufficiently well in the places where “reconciliation” is read in the AV (Lev. 6:3030And no sin offering, whereof any of the blood is brought into the tabernacle of the congregation to reconcile withal in the holy place, shall be eaten: it shall be burnt in the fire. (Leviticus 6:30); Lev. 8:1515And he slew it; and Moses took the blood, and put it upon the horns of the altar round about with his finger, and purified the altar, and poured the blood at the bottom of the altar, and sanctified it, to make reconciliation upon it. (Leviticus 8:15); Lev. 16:2020And when he hath made an end of reconciling the holy place, and the tabernacle of the congregation, and the altar, he shall bring the live goat: (Leviticus 16:20); Ezek. 45:15,17,2015And one lamb out of the flock, out of two hundred, out of the fat pastures of Israel; for a meat offering, and for a burnt offering, and for peace offerings, to make reconciliation for them, saith the Lord God. (Ezekiel 45:15)
17And it shall be the prince's part to give burnt offerings, and meat offerings, and drink offerings, in the feasts, and in the new moons, and in the sabbaths, in all solemnities of the house of Israel: he shall prepare the sin offering, and the meat offering, and the burnt offering, and the peace offerings, to make reconciliation for the house of Israel. (Ezekiel 45:17)
20And so thou shalt do the seventh day of the month for every one that erreth, and for him that is simple: so shall ye reconcile the house. (Ezekiel 45:20); Dan. 9:2424Seventy weeks are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy. (Daniel 9:24)). In the New Testament the last clause of Hebrews 2:1717Wherefore in all things it behoved him to be made like unto his brethren, that he might be a merciful and faithful high priest in things pertaining to God, to make reconciliation for the sins of the people. (Hebrews 2:17) should be translated “to make ‘propitiation’ for the sins of the people.” Elsewhere the word translated “reconciliation” is καταλλαγἠ, and kindred words, signifying “a thorough change.”
By the death of the Lord Jesus on the cross, God annulled in grace the distance which sin had brought in between Himself and man, in order that all things might, through Christ, be presented agreeably to Himself. Believers are already reconciled, through Christ’s death, to be presented holy, unblameable, and unreproveable (a new creation). God was in Christ, when Christ was on earth, reconciling the world unto Himself, not imputing unto them their trespasses; but now that the love of God has been fully revealed in the cross, the testimony has gone out worldwide, beseeching men to be reconciled to God (2 Cor. 5:19-2019To wit, that God was in Christ, reconciling the world unto himself, not imputing their trespasses unto them; and hath committed unto us the word of reconciliation. 20Now then we are ambassadors for Christ, as though God did beseech you by us: we pray you in Christ's stead, be ye reconciled to God. (2 Corinthians 5:19‑20)). The end is that God may have His pleasure in man.

“Atonement, Day Of” From Concise Bible Dictionary:

This was one of the most solemn days in the whole year, being, in common with the Sabbath, the only occasion on which the people were commanded to cease from work entirely. On the day of atonement they were also to afflict their souls, and that by a statute forever (Lev. 16:29-3129And this shall be a statute for ever unto you: that in the seventh month, on the tenth day of the month, ye shall afflict your souls, and do no work at all, whether it be one of your own country, or a stranger that sojourneth among you: 30For on that day shall the priest make an atonement for you, to cleanse you, that ye may be clean from all your sins before the Lord. 31It shall be a sabbath of rest unto you, and ye shall afflict your souls, by a statute for ever. (Leviticus 16:29‑31)). The time of the year in which the day fell — the tenth day of the seventh month — is very significant, especially when viewed in connection with the other feasts. See FEASTS.
The rites prescribed for the Day of Atonement are given (Lev. 16; Lev. 23:26-3226And the Lord spake unto Moses, saying, 27Also on the tenth day of this seventh month there shall be a day of atonement: it shall be an holy convocation unto you; and ye shall afflict your souls, and offer an offering made by fire unto the Lord. 28And ye shall do no work in that same day: for it is a day of atonement, to make an atonement for you before the Lord your God. 29For whatsoever soul it be that shall not be afflicted in that same day, he shall be cut off from among his people. 30And whatsoever soul it be that doeth any work in that same day, the same soul will I destroy from among his people. 31Ye shall do no manner of work: it shall be a statute for ever throughout your generations in all your dwellings. 32It shall be unto you a sabbath of rest, and ye shall afflict your souls: in the ninth day of the month at even, from even unto even, shall ye celebrate your sabbath. (Leviticus 23:26‑32); Num. 29:7-117And ye shall have on the tenth day of this seventh month an holy convocation; and ye shall afflict your souls: ye shall not do any work therein: 8But ye shall offer a burnt offering unto the Lord for a sweet savor; one young bullock, one ram, and seven lambs of the first year; they shall be unto you without blemish: 9And their meat offering shall be of flour mingled with oil, three tenth deals to a bullock, and two tenth deals to one ram, 10A several tenth deal for one lamb, throughout the seven lambs: 11One kid of the goats for a sin offering; beside the sin offering of atonement, and the continual burnt offering, and the meat offering of it, and their drink offerings. (Numbers 29:7‑11)). In the first we have a detailed account of the peculiar work appointed for the high priest on that day; in the second, we learn what had been shortly expressed in Leviticus 17, how the people should comport themselves on that day; and in the third we are told of certain sacrifices which were to be offered up besides those spoken of in Leviticus 16. In this passage we learn that the Day of Atonement was a gracious provision in order that the relationship of the people with the holy God who dwelt among them might be maintained.
The points to be noticed are
2. Aaron must offer sacrifices for himself and his house; a young bullock for a sin offering and a ram for a burnt offering. Aaron and his sons represent the saints who now form the church as a company of priests, and were thus, in the type, distinct from the people (Israel) as an earthly company who formed the camp.
3. For the congregation two goats were taken for a sin offering, and a ram for a burnt offering. On the two goats the lot was cast, and the one on whom the lot fell was for Jehovah, and was offered as a sin offering. This, as with the bullock for Aaron and his house, was the atonement offering Godward. The other, after being presented before the Lord, was brought forth: on him Aaron laid both his hands and confessed over him “all the iniquities of the children of Israel and all their transgressions in all their sins, putting them upon the head of the goat,” (Lev. 16:2121And Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel, and all their transgressions in all their sins, putting them upon the head of the goat, and shall send him away by the hand of a fit man into the wilderness: (Leviticus 16:21)), which was then sent away into the wilderness, a land of forgetfulness. In the two goats we have the two sides of atonement, namely, that which meets the character and holiness of God, and that which meets the need of the sinner as to the removal of his sins. (See ATONEMENT.)
4. Atonement was made for the holy place, for the tabernacle, and for the altar, because of the uncleanness of the children of Israel.
The day of Atonement being once a year — once every year, by a perpetual statute — stands in strong contrast to the one perfect sacrifice of the Lord Jesus, by which the believer is perfected in perpetuity. (See Heb. 10:1-181For the law having a shadow of good things to come, and not the very image of the things, can never with those sacrifices which they offered year by year continually make the comers thereunto perfect. 2For then would they not have ceased to be offered? because that the worshippers once purged should have had no more conscience of sins. 3But in those sacrifices there is a remembrance again made of sins every year. 4For it is not possible that the blood of bulls and of goats should take away sins. 5Wherefore when he cometh into the world, he saith, Sacrifice and offering thou wouldest not, but a body hast thou prepared me: 6In burnt offerings and sacrifices for sin thou hast had no pleasure. 7Then said I, Lo, I come (in the volume of the book it is written of me,) to do thy will, O God. 8Above when he said, Sacrifice and offering and burnt offerings and offering for sin thou wouldest not, neither hadst pleasure therein; which are offered by the law; 9Then said he, Lo, I come to do thy will, O God. He taketh away the first, that he may establish the second. 10By the which will we are sanctified through the offering of the body of Jesus Christ once for all. 11And every priest standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins: 12But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God; 13From henceforth expecting till his enemies be made his footstool. 14For by one offering he hath perfected for ever them that are sanctified. 15Whereof the Holy Ghost also is a witness to us: for after that he had said before, 16This is the covenant that I will make with them after those days, saith the Lord, I will put my laws into their hearts, and in their minds will I write them; 17And their sins and iniquities will I remember no more. 18Now where remission of these is, there is no more offering for sin. (Hebrews 10:1‑18)).

Strong’s Dictionary of Hebrew Words:

Number:
3722 (find all occurrences in KJV Bible)
Transliteration:
kaphar
Phonic:
kaw-far’
Meaning:
a primitive root; to cover (specifically with bitumen); figuratively, to expiate or condone, to placate or cancel
KJV Usage:
appease, make (an atonement, cleanse, disannul, forgive, be merciful, pacify, pardon, purge (away), put off, (make) reconcile(-liation)