The Greek word for redemption is apolytrōsis — release obtained by the payment of a ransom. It appears nine times in the New Testament and every time it carries the commercial imagery of the slave market. You were not repaired. You were purchased. Ephesians 1:7 puts it plainly: "In whom we have redemption through his blood, the forgiveness of sins." Redemption and forgiveness are not synonyms here — redemption is the transaction, forgiveness is one of its effects.
Colossians 1:13–14 adds the relocation language: "Who hath delivered us from the power of darkness, and hath translated us into the kingdom of his dear Son: in whom we have redemption through his blood, even the forgiveness of sins." The word "translated" is methistēmi — to remove from one place and establish in another. Redemption does not just clean up where you were. It moves you. You are no longer in the same domain.
Psalm 103:4 frames redemption as one item in a list of benefits that belong to the redeemed: "Who redeemeth thy life from destruction; who crowneth thee with lovingkindness and tender mercies." The Hebrew word ga'al — used here for redeem — is the same word used for the kinsman-redeemer in the book of Ruth. It implies a family obligation. God does not redeem you as a stranger buys an unknown slave. He redeems you as a family member reclaiming one of his own.
Commentary is from a charismatic Protestant perspective, drawing on KJV text and public-domain sources including Spurgeon, Andrew Murray, and Matthew Henry.