What are “ages” in Scripture?

I see many Scriptures that talk about this "age" and the "age to come." What are ages and how many are there?

Regarding the ages mentioned in Scripture, there are at least three and possibly four ages referenced in Scripture.

First, there is the age prior to Christ's appearing. Peter describes this age of waiting for Christ in the introduction to his first letter (1Peter 3-12):

Blessed be the God and Father of our Lord Jesus Christ, who according to His great mercy has caused us to be born again to a living hope through the resurrection of Jesus Christ from the dead, to obtain an inheritance which is imperishable and undefiled and will not fade away, reserved in heaven for you, who are protected by the power of God through faith for a salvation ready to be revealed in the last time. In this you greatly rejoice, even though now for a little while, if necessary, you have been distressed by various trials, so that the proof of your faith, being more precious than gold which is perishable, even though tested by fire, may be found to result in praise and glory and honor at the revelation of Jesus Christ; and though you have not seen Him, you love Him, and though you do not see Him now, but believe in Him, you greatly rejoice with joy inexpressible and full of glory, obtaining as the outcome of your faith the salvation of your souls. As to this salvation, the prophets who prophesied of the grace that would come to you made careful searches and inquiries, seeking to know what person or time the Spirit of Christ within them was indicating as He predicted the sufferings of Christ and the glories to follow. It was revealed to them that they were not serving themselves, but you, in these things which now have been announced to you through those who preached the gospel to you by the Holy Spirit sent from heaven — things into which angels long to look.

In verse 5, Peter refers to our salvation in Christ has been revealed in the "last time." In the Greek, the words are "eschatos kairos," which translated literally means "the end of the age." The salvation we have now in Christ revealed is a salvation revealed in the last age. So, ages are measured, according to Scripture, with respect to God's plan of redemption.

The age before Christ's revealing is the first age, which then gave way to our current age called "the last times" or the Church age. Notice that Peter says in verses 10-12 that the prophets were serving us in that earlier age by revealing the truth concerning Christ and the age to come, things that angels long to look upon. Paul echoes this in his letter to the Colossians (Col 1:25-27):

Of this church I was made a minister according to the stewardship from God bestowed on me for your benefit, so that I might fully carry out the preaching of the word of God, that is, the mystery which has been hidden from the past ages and generations, but has now been manifested to His saints, to whom God willed to make known what is the riches of the glory of this mystery among the Gentiles, which is Christ in you, the hope of glory.

Other verses in the New Testament (including the ones you quoted) speak of yet another age to follow this one. Again, ages are defined according to events in God's plan of redemption, so this future age will complete God's plan of redemption. According to Scripture, we know this future age is the age of Christ's kingdom on Earth, the Millennial reign of Christ described in Revelation 20. This age will, in turn, give way to a fourth age called the Eternal order. Paul describes this age in 1Corinthians 15:25-28:

For He must reign until He has put all His enemies under His feet. The last enemy that will be abolished is death. For HE HAS PUT ALL THINGS IN SUBJECTION UNDER HIS FEET. But when He says, “All things are put in subjection,” it is evident that He is excepted who put all things in subjection to Him. When all things are subjected to Him, then the Son Himself also will be subjected to the One who subjected all things to Him, so that God may be all in all.

Paul gives us glimpse of what follows the 1,000-year reign of Christ, when he describes the age to follow as a time when God becomes "all in all." John also gives a tantilizing description of this future age in Revelation 21 and 22.