I've been pondering this one a lot recently (as a Christian). On thing, Griphook - as you may have noticed, there's not one single answer to your question!
Different people interpret the death of Jesus in different ways, and have different ideas of what sin is and how God deals with it.
AMIS - your answer is soooo Book of Hebrews, which I absolutely love. Love it, love it, love it.
I am thinking more and more along the lines of defining sin as violence - when we sin, we do / say / think violent things that damage ourselves / others / our world. And we can't stop ourselve from saying / thinking / doing these things, however much we hate them, because we are human and part of a competitive ecosystem in which each living thing has to fight for survival, at the expense of the other (plants compete for soil's nutrients, animals for food, humans for significance / affection / money etc).
So Jesus' death, as both fully God and fully human, was a voluntary act of non-violence, laying down of rights and submitting to one of the most horrifically violent deaths the world had yet devised. One of the Old Testament passages which has been used since very early Christianity to talk about Jesus' death says that he was 'like a lamb before the slaughter'. And the resurrection is the overcoming of the the violence of the world's system and the triumph of God's love that heals all damage and violence, and enables us to be changed (to repent) and through the Spirit of God be made into those who can work and live for peace and truly live for others rather than being condemed to compete against them.
That's how I see it anyway. There's so much richness and mystery in the death and rising of Jesus, I'm not sure we'll ever get to the bottom of it.