The Atonement of Christ is one of the most important and rich concepts in the entire Christian faith. Why, exactly, did Jesus go to the cross, and what did it actually achieve? Throughout history, a number of weird and wonderful theories have been proposed. Some of these are false, some are partially true but don't give the full picture, and one stands out as the greatest and most Biblically based. Let's look at a few of these theories of atonement.
Ransom Theory
Ransom Theory contends that Christ's death was payment, as ransom, to Satan for our souls. This unsubstantiated theory contends that man is in bondage to Satan, and that by dying, Christ broke that bondage. This theory gives way too much credit to the devil, not only overlooking the fact that God owes him absolutely nothing, but also that we don't either. Rather, it is God and His justice that demand death as penalty for sin. Satan has no authority at all, and given that he, himself, is a sinner, and no atonement has been made for him, he will actually suffer the same (and indeed, worse) punishment as the impenitent. Ransom Theory, therefore, is the equivalent of handing your bail money over to a death row inmate. To be clear, a ransom has effectively been paid, but not to Satan. It is God to whom we owe our lives, and we have been set free from His wrath, which we were justly owed.
Dramatic Theory
Similar to Ransom Theory, Dramatic Theory suggests that the cross was the culmination of an ongoing battle between good and evil, ensuring God's victory over Satan. Although it's true that the cross overcame sin, and Christ did crush the devil's head with it as he struck His heel, this theory falls just short because it fails to explain how, exactly, this is the case.
Recapitulation Theory
Recapitulation Theory suggests Christ's life recapitulated all stages of human life, thus reversing the course of human nature from disobedience to obedience. This theory is supported by no Scriptural evidence.
Moral Influence Theory/Example Theory
Moral Influence Theory posits that Christ's sacrifice is a show of love that moves man to repentance. It sees man as spiritually sick and in need of a softened heart, brought about by seeing God's love for us. It is actually true that God shows His love for us in that while we were sinners, Christ died for us (Romans 5:8) and that we love Him because He first loved us (1 John 4:19). Example Theory, similarly, postulates Christ's sacrifice is simply an example for us to follow. Just as Jesus submitted Himself to God and obeyed, so also must we do so. The difference between the two is that Moral Influence Theory is about Christ showing us how God loves us and causes us to love Him in turn, whereas Example Theory is about Christ showing us how to submit to Him in faith and obedience.
These two theories both have the same shortcoming: They overlook why Christ went to the cross. It is absolutely true that He did so in love and obedience, but going to the cross is not a show of love unless there is a point to it. The only way Christ's sacrifice could be a show of love is if it actually achieves something for the objects of that love, namely us.
Penal Substitutionary Atonement
While a lot of the other theories do have some merits, they all fall short in that they overlook God's justice. They all miss the fact that God, not Satan, requires payment for sin, and that payment is death. God shows His love for us not by randomly killing Jesus, but by punishing Him in our place. This does give us an example, not only of how to love God, but even each other (and in particular, how husbands are to love their wives, Ephesians 5:25). However, it is not just a show of love or an example, as unless something is achieved, it does neither. It is foolish to commit suicide just to show you love someone, it is truly loving to lay down your own life so your loved ones may keep theirs. This is, indeed, what Christ does, as He, who knows no sin, became sin for us, so that we may become the righteousness of God in Him (2 Corinthians 5:21).
Of all theories of atonement, Penal Substitutionary Atonement is the most robust. It adequately explains why Jesus went to the cross, what it achieved, and how it achieved it. This does not rule out other theories, but without PSA, these theories all fall short, giving, at best, a partial view, but by no means a complete one.