Generally today, most western Churches (ie Catholic and Protestant), as far as I'm aware, tend to hold to the idea of Substitutionary Atonement: Jesus had to die to take the punishment (from God, because God is so just that he couldn't not do punishment) for mankind's sins. But I read that, among theologians, this is an idea that became popular in the latter middle ages, and before that, in the early middle ages, a more popular theory was Ransom Theory: Jesus died to, with his death, buy souls from Satan. But then God tricked Satan by resurrecting Jesus.
Is this correct? If so, what caused the shift? Are there any other theories that were popular or influential among theologians? Did eastern Churches have different ideas?
A crash course in historical views of the death of Jesus (aka the atonement).
Partly the issue is generated by the New Testament texts themselves, which speak of its significance (directly, or in texts that came to be understood to refer to Jesus' death) in multiple ways. This includes removing the punishment for sin (expiation), turning aside God's wrath against sin (propitiation), purchasing back people from slavery (redemption), returning enemies to a state of friendship with God (reconciliation), defeating Satan and death (victory), acting as a model for Christian faithfulness and suffering (exemplary), and imputing Christ's righteousness to believers (exchange).
All these aspects or facets can be found in a number of texts, and serve as the generative 'ground' for later theological reflection on Jesus' death. And good examples can be found of most of these motifs being taken up by early church theologians.
For example, the ransom theory is develop by Origen, who viewed it in terms of all people, due to sin, rightfully belonging to Satan. Thus, the ransom as paid to Satan to free people from his dominion. Yet Satan could not hold onto Jesus, and so lost both his slaves and the ransom itself.
In terms of western theology, the consolidation of the Latin-Western tradition and the move into the medieval period sees a major development in Anselm of Canterbury (d. 1109), whose most famous work is Cur Deus Homo (Why God became man), in which he develops a satisfaction theory, influenced in part probably by society in which he lived, which demanded restitution of honour for crimes committed. God, then, having his honour diminished by sin against him, requires restitution in forgiving sins, which means either sinners must do so, or Christ must do so in their place. And, yes, sinners cannot do so, since anything they would offer would alreaady be owed to God, nor can they offer anything sufficient. Anselm's view was influential though not exclusively adopted.
Penal Substitionary atonement in its more modern forms is really a development linked to the Reformation, altering the place of honour/restitution in Anselm's theory, to focus on the justice of God and thus the necessity of punishment of sin. It is this, in Protestant thought, which Christ substitutes and provides. This is pretty clear in Luther, and moreso in Calvin I'd say.
Roman Catholic theology rejects Penal Substitionary atonement, though that's not always clear to everyone. Specifically, it rejects the idea that the death of Christ is a punishment. It tends, instead, to develop along the lines from Anselm, and Aquinas, to view the atonement as a matter of satisfaction, not of justice/punishment, but also emphasising and highlighting Jesus' death as an act of love and self-sacrifice.
Eastern Orthodox theology focuses its attention differently - Jesus' death is generally seen more in terms of the defeat of sin, Satan, and death, and its effect for believers is tied up with being united in Christ with the divine nature, so that believers are participating in the divine nature and being transformed by that participation.
This is a fairly 'bird's eye view' overview of 2000+ years of theology, so I am more than open to some follow-up questions.