Throughout the history of Christian thought there have been a variety of models for explaining the mechanics of the Atonement. How exactly does it work, and why did Jesus have to suffer? Most Christians (including Latter-day Saints) seem content to leave the mechanics of the Atonement a mystery, which is fine. But for those interested in pursuing the question, here is a summary of the three traditional theories of atonement and their variations throughout Christian history. This is taken straight from Wikipedia (no need to reinvent the wheel!).
Ransom Theory: The first metaphor, epitomised by the “ransom to Satan” theory, was used by the fourth-century theologian Gregory of Nyssa based on verses such as Mark 10:45 – “the Son of Man came … to give his life as a ransom for the many”. In this metaphor Jesus liberates mankind from slavery to Satan and thus death by giving his own life as a ransom. Victory over Satan consists of swapping the life of the perfect (Jesus), for the lives of the imperfect (mankind).
Variation 1: A variation of this view is known as the “Christus Victor” theory, and sees Jesus not used as a ransom but rather defeating Satan in a spiritual battle and thus freeing enslaved mankind by defeating the captor.
Satisfaction Theory: The second metaphor, used by the eleventh century theologian Anselm, is called the “satisfaction” theory. In this picture mankind owes a debt not to Satan, but to sovereign God himself. A sovereign may well be able to forgive an insult or an injury in his private capacity, but because he is a sovereign he cannot if the state has been dishonoured. Anselm argued that the insult given to God is so great that only a perfect sacrifice could satisfy and Jesus, being both God and man, was this perfect sacrifice.
Variation 1: A variation on this theory is the commonly held Protestant “penal substitution theory,” which instead of considering sin as an affront to God’s honour, sees sin as the breaking of God’s moral law. Placing a particular emphasis on Romans 6:23 (the wages of sin is death), penal substitution sees sinful man as being subject to God’s wrath with the essence of Jesus’ saving work being his substitution in the sinner’s place, bearing the curse in the place of man (Galatians 3:13).
Variation 2: Another variation that also falls within this metaphor is Hugo Grotius’ “governmental theory”, which sees Jesus receiving a punishment as a public example of the lengths to which God will go to uphold the moral order.
Moral Influence Theory: The third metaphor is that of healing, associated with Pierre Abélard in the eleventh century, and Paul Tillich in the twentieth. In this picture Jesus’ death on the cross demonstrates the extent of God’s love for us, and moved by this great act of love humankind responds and is transformed by the power of the Holy Spirit. This view is favoured by most liberal theologians as the moral influence view, and also forms the basis for Rene Girard’s “mimetic desire” theory (not to be confused with meme theory).
In the Latter-day Saint tradition a few attempts have been made by LDS thinkers to formulate a new theory of Atonement that incorporates LDS thought and scripture. Notable examples are Cleon Skousen’s “intelligence theory”, and Blake Ostler’s “compassion theory” of the Atonement. But, for the most part, Latter-day Saints simply borrow language from all 4 of the above theories and do not have a single unique concept of the mechanics of the Atonement (at least that is my experience). There are various analogies that have become popular among the Latter-day Saints, including President Packer’s “mediator” analogy and Stephen Robinson’s “bicycle” analogy. But these analogies don’t seem to propose a new theory so much as they try to explain an existing theory.