Redemption is a metaphor before it is a doctrine, and the metaphor is legal and commercial: to redeem is to buy back, to secure release by payment. The Hebrew vocabulary is entirely concrete. The go'el is the kinsman-redeemer, the relative obliged to buy back a family member's land or person when they have fallen into debt or servitude (Leviticus 25:25-55), the institution that supplies the whole plot of the book of Ruth. Padah is the ransoming of a firstborn or a slave. From this domestic legal machinery the term is transferred to God, who "redeemed" Israel from Egypt (Exodus 6:6, Deuteronomy 7:8) and whom Isaiah repeatedly calls Israel's redeemer.
Transferring the metaphor raises the question every tradition that uses it has had to face: if a price is paid, to whom is it paid. Early Christian writers who pressed the ransom image, notably Origen and Gregory of Nyssa, answered that it was paid to the devil, who had acquired a claim over humanity, Gregory adding the image of the divinity concealed in the humanity as a hook concealed in bait. Gregory of Nazianzus objected sharply that the devil is owed nothing whatever and that paying him would be an outrage, and Anselm's Cur Deus Homo abandoned the scheme altogether in favour of satisfaction owed to God. The exchange is a good illustration of how a metaphor, pressed for consistency, generates doctrine and then generates objections to itself.
In Judaism, geulah remains chiefly corporate and historical. The Exodus is its paradigm, the return from exile its continuation, and the messianic age its completion, and the Passover Haggadah keeps the paradigm operative by requiring each person in every generation to see themselves as having personally come out of Egypt. Alongside this stands pidyon shevuyim, the redemption of captives, a major halakhic obligation with a long and grim practical history, and with its own rule, much discussed in modern Israeli debates over prisoner exchanges, that captives are not to be ransomed for more than their worth, lest further captures be encouraged.
In Christianity, redemption became one of several overlapping terms, with justification, reconciliation and atonement, for what Christ's death and resurrection accomplish, and the New Testament uses the purchase language directly: "you were bought with a price" (1 Corinthians 6:20). Modern theology has questioned whether transactional language distorts what it is used to describe, and liberation theologies have pushed the Exodus sense back to the front, reading redemption as historical liberation from oppression as well as deliverance from sin, a reading contested by those who regard it as reducing the doctrine to politics.