To redeem means to get possession of something you don't own by paying for it. Perhaps ownership of something was lost and you want to regain it. There is the case where a man was "so poor . . . as to sell himself" (Lev 25:47) as a slave. Fortunately "one of his brothers may redeem him" (:48) or "if he prospers, he may redeem himself" (:49). "If a ransom is demanded of him, then he shall give for the redemption of his life whatever is demanded of him" (Ex 21:30). In man's case we had been "severed from Christ" (Gal 5:4) and "alienated and hostile in mind" (Col 1:21). "God created man in His own image" (Ge 1:27). But now Adam had "a son in his own likeness" (5:3), and due to the fall of man, this image was not equivalent to the original likeness because man's relationship with God had deteriorated. Unfortunately man was caught in a "yoke of slavery" (Gal 5:1) and "through fear of death were subject to slavery all their lives" (Heb 2:15). "'We are slaves; yet in our bondage, [and] our God has not forsaken us, but has extended lovingkindness to us'" (Ez 9:9). "God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us" (Ro 5:8). Christ had to accomplish this because "no man can by any means redeem his brother or give God a ransom for him--for the redemption of his soul is costly, and he should cease trying forever" (Ps 4:7-8). Redemption means to buy back by making payment, and in man's predicament, it pertains to freeing him from his bondage to sin. Therefore Jesus explained, "'the Son of Man did not come to be served, but to serve, and give His life a ransom for many'" (Mk 10:45). The payment of a ransom applies to a loosing or setting someone free, and by this, the Father through Christ is regaining possession of something that originally was his as a "redemption of God's own possession" (Eph 1:14). It is "the church of God which He purchases with His own blood" (Ac 20:28). "Without shedding of blood there is no forgiveness" (Heb 9:22). Christ "gave Himself as a ransom for all" (1Ti 2:5) since "God was in Christ reconciling the world to Himself" (2Co 5:19). Someone once told me that it means everyone automatically goes to heaven. But you must appropriate what Christ has done to make it personally official.
Christ "cancelled out the certificate of debt consisting of decrees against us; and . . . nailed it to the cross" (Col 2:14). However, "the word of the cross is to those who are perishing foolishness" (1Co 1:18). But it is serious and not just academic. Christ "redeemed us . . . having become a curse for us" (Gal 3:13) by his crucifixion. It was from "the curse of the Law" (:13) for "forgiveness of our trespasses" (Eph 1:7). It is "redemption [for] the forgiveness of sins" (Col 1:14) and "transgressions that were committed under the first covenant" (Heb 9:15). It was to redeem us "from your futile way of life inherited from your forefathers" (1Pe 1:18) and "from every lawless deed . . . [to be] zealous for good deeds" (Titus 2:14). He "rescued us from the domain of darkness" (:13) and "from the hand of our enemies" (Lk 1:74). It was a "testimony given at the proper time" (1Ti 2:6) and "displayed publicly as an atonement in His blood through faith" (Ro 3:25). "Jesus said, "'This cup is the new covenant in My blood'" (1Co 11:25) "poured out for many" (Mk 14:24). Redemption has numerous important objectives. One result is that we are "justified as a gift by His grace" (Ro 3:24). It is so "we might receive the adoption as sons" (Gal 4:5) and to "receive the promise of eternal inheritanace" (Heb 9:15). We have "the first fruits of the Spirit" (Ro 8:23) and are "sealed" (Eph 4:30) by the Holy Spirit. Christ to us becomes "wisdom from God, and righteousness, and sanctification" (1Co 1:30). Through Christ we "are believers in God . . . so that your faith and hope are in God" (1Pe 1:21). "Through His own blood, He entered the holy place once for all" (Heb 9:12) having "purchased for God . . . men from every tribe and tongue and people and nation" (Rev 5:9). "Do you not know . . . that you are not your own?" (1Co 6:19,20). "You were bought with a price; do not become slaves of men" (7:23).