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

“Forgiveness” From Concise Bible Dictionary:

There are three Hebrew words translated to forgive.
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.

“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.

Strong’s Dictionary of Hebrew Words:

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)