That Christ died for our sins is not in dispute among Christians. It is in the earliest Paul quotes, it is in every historic , and no tradition here questions it.
The question is what happened, mechanically. The New Testament reaches for many pictures and never systematizes them: a ransom paid, a sacrifice offered, a legal verdict satisfied, a victory won, a debt cancelled, an example set, a family reconciled. Theologians have taken different pictures as the central one and read the rest through it.
Most Christians hold several of these at once. The disagreement is usually about which is load-bearing, and about whether one of them says something about God that the others would not.
Isaiah 53, Romans 3:21-26, Colossians 2:13-15
21But now apart from the law, a righteousness of God has been revealed, being testified by the law and the prophets; 22even the righteousness of God through faith in Jesus Christ to all and on all those who believe. For there is no distinction, 23for all have sinned, and fall short of the glory of God; 24being justified freely by his grace through the redemption that is in Christ Jesus, 25whom God sent to be an atoning sacrifice through faith in his blood, for a demonstration of his righteousness through the passing over of prior sins, in God's forbearance; 26to demonstrate his righteousness at this present time, that he might himself be just and the justifier of him who has faith in Jesus.
All these views hold that Jesus died and rose, that his death was for us and not merely an example of unjust suffering, and that reconciliation with God comes through it rather than through our own achievement. They are readings of one event, not competing events.
Penal substitution
Sin earns a penalty, and Christ bore that penalty in the place of sinners, so that God can forgive without setting justice aside.
What it claims
God's justice is not a mood but part of his character, so sin cannot simply be waived. At the cross Christ stands in the place of the guilty and receives what their sin deserved. God is therefore both just and the one who justifies, which is exactly the pair of things Paul says is demonstrated at the cross.
Who holds it
The dominant view in Reformed and evangelical Protestantism, stated in the Reformation confessions and taught across Baptist, , Anglican evangelical, and most non-denominational churches. Anselm's satisfaction theory is its medieval ancestor, and Calvin gave it its familiar legal shape.
The texts it stands on
Isaiah 53:5-6 Pierced for our transgressions, and the LORD laid on him the iniquity of us all.
Romans 3:25-26 Set forth as a propitiation, so that God might be just and the justifier of the one who has faith in Jesus.
2 Corinthians 5:21 He made him who knew no sin to be sin on our behalf.
Strongest case
It takes the language of guilt, wrath, and at face value, and it explains why the cross was necessary rather than merely moving. It also gives a direct answer to the question that troubles most people who are actually afraid: what happens to what I have done.
Hardest question
Critics argue it can be told in a way that pits the Father against the Son, as though a kind Jesus rescues us from an angry God, which every careful advocate denies but which popular presentations often imply. Others object that punishing an innocent party is not obviously justice at all.
Holding the map
It is worth noticing that the New Testament itself never picks one. Within a single letter Paul can write in courtroom language, in slave-market language, in temple language, and in battlefield language, apparently without feeling the strain.
Most traditions officially hold more than one of these. The Reformed confessions that state penal substitution also affirm Christ's victory; Orthodox that sings of trampling down death also speaks of sacrifice. The real question in most arguments is not which is true but which is central, and what is lost when a church only ever preaches one.
Questions that press on this one
None of these stands on its own. What you conclude here narrows what you can say there.
