Concise Bible Dictionary:
A period of time marked off by that which distinguishes it from other periods, as the Patriarchal age, the Apostolic age. Though these terms do not occur in scripture, many allusions are made to the different ages. Thus we speak of the Antediluvian age, and scripture speaks of the same period as the “old world,” for example, the ordered scene which God did not spare (2 Peter 2:5). The “Patriarchal age” embraces the time from the call of Abraham to the release from Egypt and the giving of the law. “From Adam to Moses” excluding both, is an epoch when men’s sins could not be classed as transgressions, seeing there was no definite law such as was given to Adam, or such as was administered by Moses (Rom. 5:13-14). Again, from Moses to Christ formed a definite period: “the law was given by Moses, but grace and truth came by Jesus Christ” (John 1:17); “the law and the prophets were until John: since that time the kingdom of God is preached, and every man presseth into it” (Luke 16:16). The time when Christ was on earth is in John’s first epistle repeatedly called “the beginning”: the beginning of Christianity, though it is not called an age. We may also distinguish the period of the Church (from the day of Pentecost until the Rapture of the saints), though it is separated from recognized times and seasons. Our Lord not only spoke of the age in which He was on earth, but referred some events to the end or “completion of the age” (Matt. 13:39-40, 49). The disciples also asked what would be the sign of that completion (Matt. 24:3). Our Lord also spoke of the “coming age,” when His own should receive life eternal (Luke 18:30; compare also Matt. 12:32; Eph. 1:21; Eph. 2:7). God’s kingdom is a “kingdom of all ages” (Psa. 145:13, margin). Eternity, in reference to the glory of God, to the blessing of the saved, and to the punishment of the wicked, is again and again called the “age of ages,” often translated “forever and ever”: all being consummated in the eternal state. The Greek word is αίών throughout, though often translated “world” in the AV (KJV).
From Anstey’s Doctrinal Definitions:
A term referring to an epoch or period of time that has run, or is running, or will run, its course on earth. Such periods are referred to as "the ages of time" (2 Tim. 1:9; Titus 1:2).
The Lord spoke of two ages in particular in His earthly ministry: "this age" and "the age to come" (Matt. 12:32). "This age" is the Mosaic age, which began at Sinai in the giving of the Law, and was in progress at the time of the Lord's first advent. When He was rejected and cast out of this world, this age received a new characterization, and is now called "the present evil age" (Gal. 1:4). This is on account of the "princes of this age" committing the greatest of all sins—the crucifying of the Lord of glory (1 Cor. 2:6, 8). “The age to come” is the Millennium, the 1000-year public reign of Christ, yet to come in the ways of God (Mark 10:30; Eph. 1:21; Heb. 2:5; 6:5: Rev. 20:4). There have been “other ages” before the Mosaic age that have already run their course, as Paul indicates in Ephesians 3:5—such as: the Antediluvian age, the Patriarchal age, etc.
Some have thought that the present calling of God by the gospel has put the Mosaic age in abeyance, and the ages of time will not begin again until some future day. But, this is not true; the Mosaic age is still running its course on earth today. The coming of the Holy Spirit and the introduction of Christianity did not bring it to an end; nor did it begin a new age. However, while the Mosaic age is not suspended, God's formal connection with Israel as a nation is in abeyance. Those who believe the gospel preached today are called out from the Jews and the Gentiles to be part of the Church of God. They are delivered "out of this present evil age" and are no longer part of it, as far as their position is concerned (Gal. 1:4). The Church, therefore, has no connection with the earth and the periods of time. Hence, to speak of this present time when the gospel of God’s grace is going out to the world as "the Church age" is not doctrinally accurate. Christians still must walk through this present evil age, but positionally they are not part of it. And sad to report, some Christians today are turning from their steadfastness in following Christ and are loving "this present age" and are settling down in the world as a result. Demas is an example (2 Tim. 4:10).
This present age is under the control of Satan who is its god and prince (2 Cor. 4:4; Eph. 2:2), and it is going on to judgment. We know from the prophetic Scriptures that it has at least seven more years remaining to it, which will run their course after the Church is called to heaven at the Rapture. These seven years correspond with the seventieth week of Daniel (Dan. 9:27). This age will close at the Appearing of Christ in what is called, "the completion of the age" (Matt. 13:39-40, 49; 24:3; 28:20). At that time, the Lord will bring in "the age to come," the Millennium (Matt. 12:32; Mark 10:30; Eph. 1:21; Heb. 2:5; 6:5). When the Millennium has run its course of 1000 years, the Eternal State will begin (Rev. 21:1-8). Scripture calls this, "the ages of ages" (Gal. 1:5; Eph. 2:7; 3:21; 1 Tim. 1:17; 1 Peter 5:11; Rev. 5:13; 22:5). Technically speaking, it is not really an age, because ages have to do with time, and there is no time in eternity.