While as a Jew I do not subscribe to this doctrine, the "ransom" of Christ is his becoming the "Lamb of God," an offering to humans as "food" for their journey to "passover" from the mortal state to that of the divine.
Originally atonement theology was very basic. It suggested that God was being appeased by the suffering of his own Son. But it changed over the centuries, and currently the teaching is that Christ offered himself to humans, not to God. As 2 Peter 1:4 states, Jesus paid the price of himself in order to "enable you to share in the divine nature."
As the lamb was food for the Exodus out of Egypt for the Jews, the life of Jesus is Viaticum, food for the journey to pass from mortality to share in God's divine nature. Jesus flesh and blood are "paid" over to humans as "true food" and "true drink," not on an altar to God but on a table to humans.--John 6:55.
"Redemption" is different, however. It refers to restoring people by legal claim, or restoring something to people by legal means. Jews view the Exodus as God's redemption of his people from slavery. God paid no one to redeem the Jews as God owes no one. But God had legal claim to them by means of the Abrahamic covenant. The same covenant claimed rights and privileges to the Israelites that they could not possess as slaves in Egypt. Thus a "redemption" took place via the Exodus. Redemption in this sense involves restoration.
Some Christians still hold to an "appeasement" view of Christ's ransom. In these instances, "redemption" is somewhat of a synonym for "ransom" in a paradigm when Christ offers a sacrifice to God.
But where the theology that rejects appeasement exists, "redemption" keeps the original meaning as used by the Jews. In Christian terms it refers to restoring what was lost in Eden (according to Christianity's Original Sin doctrine), giving this back to humans via the "cost" of Christ's sacrifice.
Jehovah's Witnesses believe in appeasement theology, where God requires blood to be spilled to appease his sense of justice. This view has been rejected by almost all forms of mainstream Christianity today. Even some Fundamentalist movements find the appeasement view appalling as it draws God as a bloodthirsty deity.