Redemption is one of the central themes of the Bible. God redeemed Israel from Egypt (Ex 12), Messiah is both a ruler and redeemer (Acts 7.35), Messiah gave himself to redeem us from lawlessness (Titus 2:14), etc.
While the word redeem has several related uses in Scripture, its sense is to take posession of something that was previously owned. So in Lev 25.23ff if someone has to sell his inherited portion of the land of Israel, he or his relative has the right to redeem the land by paying the new owner for the crops the land might be expected to produce.
There is a difference between redemption and purchase. In redemption the redeemer has a prior right to something (or someone) and pays a price to get that thing back from another who has posession of it. In a purchase, there is no prior right or relationship. So, if I go to a car dealer and buy a new car, it is a purchase. If the police have towed my car and I pay to get it back, then it is a redemption. In both cases I would pay money to get the car, but it is the prior ownership that makes the transaction a redemption rather than a purchase.
Ownership is only one of several relationships that form the basis of redemption. In the case of land or home ownership in Biblical Israel, people were sojourners on property that is actually owned by God. God gives His people a perpetual right to the use of a particular piece of real estate by inheritance. So when an Israelite redeems land that is part of his inheritance, the relationship is not prior ownership, but grant of inheritance by God. Yet an Israelite can only redeem his inheritance, not another's. So there still must be a personal relationship between the redeemer and the land that is bought back.
Family relationship can also form the basis for redemption. If an Israelite sells himself into slavery, his near kinsman has the right to redeem him by buying out his contract from the master. As noted above, a kinsman also has the right to redeem a piece of family property which has been sold to someone outside the family.
Interestingly, the Hebrew for redeemer is also the word for avenger. So the near kinsman who would redeem a relative from slavery or redeem a field sold to another would also be responsible to avenge his relative's murder. All of these things are relevant to our relationship with God and with his Son.
To see how this works, consider first God's redemption of Israel from Egypt as told in Exodus. The essence of the dispute between God and Pharaoh was the question of who owned Israel. Pharaoh asserted that they were slaves and hence his property. God disagreed, asserting kinship based on His covenant with Abraham, Isaac and Jacob. In Exodus 4.22, God says, 'Israel is my firstborn son.' God then proceeds to redeem Israel from Egypt, taking vengeance on Pharaoh in the process for his mistreatment of Israel while they were slaves.
That explains the redemption of Israel, but how do we get from there to the redemption of all mankind? The answer to that goes to inheritance and kinship. In Genesis, humanity was given dominion over the earth. In much the same way as Israel was given permanent use of the Land, humanity is given permanent use of the earth. God did not relinquish ownership, just management and use of the place. That we have sold ourselves into slavery to Satan does not negate the original grant of dominion. As a man, Y'shua participates in that grant. As the Son of God, He is free from the sin that holds the rest of us in bondage. So He is entitled to step in an take possession of the Earth. Were He to have done that 2000 years ago, Israel might have been redeemed, but the rest of humanity would still have been in bondage because of the sin debt. (That, by the way, was the scenario that the Jews were expecting.) To avoid that, He paid the debt for all humanity instead of taking immediate possession during His first advent. Since the price for the sin of rebellion and adultery is death, He took those sins, died and imputed His blood to us.
The final piece of the redemption puzzle is detailed in Revelation. The seven seals describe the process whereby He proves that He is the rightful owner of the earth. Once that title is proved, the seven trumpets announce the coming of the King. At the last trumpet, the dead will be raised, we who remain will be changed and our redemption will be complete. Our near kinsman will have removed us from slavery and death. (1 Cor 15.50-58) When our redemption is done, the King (our kinsman) will take vengeance upon His enemies - this is the seven bowls.