What is redemption? It's a word we used to hear a lot in Christian circles; maybe not so much any more. Scripture talks about redemption in both the Old and New Testaments: it's worth considering what the word means.  The idea of redemption is buying back.  We might consider a few verses from Scripture:

6 They depend upon their wealth, and boast themselves in the abundance of their riches. ...
7 None can by any means redeem his brother, nor give to God a ransom for him,
8 (For the redemption of their soul is costly, and must be given up for ever,)
9 That he should still live perpetually, and not see corruption.
Psalm 49:6–7, DBY
I will hiss for them, and gather them; for I have redeemed them: and they shall multiply as they used to multiply.  Zechariah 10:8, DBY

who gave himself for us, that he might redeem us from all lawlessness, and purify to himself a peculiar people, zealous for good works.
Titus 2:14, DBY

Psalm 49 establishes the principle that redemption implies a cost.   When Scripture talks about redemption, it talks about giving up one thing for another. "[T]he redemption of [a] soul is costly, and must be given up forever." It's a purchase.

The Old Testament prophets repeatedly refer to Israel as God's redeemed people (see Isaiah 44:23; Jeremiah 31:11; Micah 6:4). In the New Testament we have redemption taught slightly differently. In the Old Testament we have God as the Redeemer; in the New we see not only Christ our Redeemer, we see something of the price that He paid. Where the Old Testament puts a focus on God the Redeemer, the New Testament spotlights the cost of redemption.

What is the cost of our redemption?  "[W]e have redemption through his blood" (Ephesians 1:7).  Our redemption came at the cost of the blood of Christ Jesus. Hebrews says, 

11 But Christ being come high priest of the good things to come, by the better and more perfect tabernacle not made with hand, (that is, not of this creation,) 12 nor by blood of goats and calves, but by his own blood, has entered in once for all into the holy of holies, having found an eternal redemption.
Hebrews 9:11–12, DBY

We're not discussing atonement right now, so we're going to skip over the really juicy bits here; but the plain teaching of Hebrews 9 is that Christ has found "eternal redemption", and it came at the cost of "his own blood" (Hebrews 9:12).

Let's take a moment and consider "eternal redemption". The epistle to the Hebrews establishes this as the main point of its doctrine. The Eternal Son came from Heaven to earth to die for us. He is our Priest, He is also our Sacrifice. Hebrews carefully points out that He cannot sacrifice Himself again (Hebrews 9:27–28). The death of Christ is not only the Great Sacrifice that all the Old Testament sacrifices pointed to in type; it was also the Final Sacrifice. It cannot be repeated, and it cannot be used up. Christ Jesus has found an eternal redemption: there is no more price to be paid. God has found what He was looking for, and He has stopped looking.

There is no more to be done, Christ has done it all. Having been redeemed, we can't be un-redeemed. Christ has found an eternal redemption: it's settled in Heaven, and cannot be changed.

We've considered Romans 3 with respect to justification, but let's consider what is says about redemption:

20 Wherefore by works of law no flesh shall be justified before him; for by law is knowledge of sin. 21 But now without law righteousness of God is manifested, borne witness to by the law and the prophets;  22 righteousness of God by faith of Jesus Christ towards all, and upon all those who believe: for there is no difference;  23 for all have sinned, and come short of the glory of God;  24 being justified freely by his grace through the redemption which is in Christ Jesus;  25 whom God has set forth a mercy-seat, through faith in his blood, for the shewing forth of his righteousness, in respect of the passing by the sins that had taken place before, through the forbearance of God; 26 for the shewing forth of his righteousness in the present time, so that he should be just, and justify him that is of the faith of Jesus.
Romans 3:20–26, DBY

There is a redemption "in Christ Jesus" (Romans 3:24). How can we get in on that? Through "faith in His blood" (Romans 3:25). How is that redemption applied to me? It is on the basis of this redemption that we are justified: God doesn't justify us only because He's loving and kind. God justifies us with a view to the redemption in Christ Jesus. God can justly justify us because Christ Jesus redeemed us with His own blood (Romans 24–26).  We are justified by faith (Romans 5:1), and that is what brings us into the good of this redemption.

Christ having redeemed us, we no longer belong to ourselves. This is the other side of redemption: redemption is buying back at a cost. Christ has redeemed us, and now we are His. Let's consider 1 Corinthians: 

19 Do ye not know that your body is the temple of the Holy Spirit which is in you, which ye have of God; and ye are not your own? 20 for ye have been bought with a price: glorify now then God in your body.
1 Corinthians 6:19–20, DBY

Ye have been bought with a price; do not be the bondmen of men.
1 Corinthians 7:23, DBY

So redemption is a practical thing: redemption means not only the Christ has bought me at a terrible price; but He now owns me by virtue of the price He paid. That means I don't have the right to live as I choose. I'm not my own, I've been bought by Someone else. 1 Corinthians develops this as the basis of practical instruction: we're not our own. We haven't the right simply to gratify our own desires.

What is redemption? It's buying back. What is eternal redemption? It's the Scriptural teaching that Christ has bought us at the cost of His own blood. He has bought us, He has paid for us, we are His.