Salvation theology is actually a different topic than atonement theology, although both are being combined in this thread.
There are a lot of views on salvation, and how we can obtain it. Some rely solely on the grace of God, and nothing we can do will achieve it. Others believe we have to simply believe in God, Jesus, and some basic tenets of Christian faith, and then we’ll be saved. Still others will say that we have to follow Jesus; that salvation is the result of a life of selflessness and sacrifice.
Atonement, on the other hand is really complicated, and I think that’s what we’re really talking about here.
While substitutionary atonement (Jesus was a perfect substitute for our sins; his death on the cross allowed God to forgive our sins) is the most popular one, there are actually two other dominant “theories” of atonement.
Substitutionary atonement is actually not the oldest view, but was formulated in the high Middle Ages by Anselm of Canterbury. This is now the official stance of the Catholic Church and in various forms, held by most evangelical and conservative Protestant churches. The reasoning is that God demands satisfaction for the sins committed against him, but nothing we can do will suffice; therefore, Christ had to die as a perfect substitute for our sins. This is often justified with appeal to the Hebrew scriptures, and considers Jesus as the perfect sacrificial lamb. As some posters have noted, however, most Jewish scholars consider this unconvincing.
The second most popular view, taught in most liberal and some moderate Protestant churches, is the “moral” theory of atonement. The idea is that Jesus, by giving himself to die on the cross, shows humanity the way of sacrifice and the way of selflessness, even unto death, is how we must live. The whole “WWJD” phenomenon, although most popular with evangelical teenagers who probably hold to the sacrificial substitutionary atonement theory, was actually the product of this liberal Christian view.
The third, and oldest, view, which is held by the Orthodox churches, is the “ransom” or “Christus victor” theory. The earliest version of this theory saw humanity as being held bondage by the devil, unable to escape because of our sinfulness. Jesus’ death was given to the devil as a ransom, so that we might be freed. Jesus’ resurrection ultimately showed that the devil did not have the last word, and that death could not contain him. Later versions deemphasized the ransom aspect because it portrayed God as striking a deal with the devil! Nowadays, many liberal Protestant churches teach a version of this combined with the moral theory, seeing Jesus’ life and death as God’s way of taking on everything the powers of earth could possibly dish out, and then emerging victorious through the resurrection, showing that we no longer need to fear following Jesus, because death and sin have been defeated.
I like the Christus victor view the most, but ultimately, all theories of atonement are the result of Christian views on why Jesus died. It was a shocking idea, and still is. If God was on Jesus’ side, then why would God allow him to be crucified? Thus, we get a lot of explanations, but none of them are perfect, and we should view them as imperfect, human rationalizations of the death of Jesus rather than concrete truth.