There are differences between the understanding of redemption in the Catholic and Orthodox churches; I can speak only for the Orthodox part, and hopefully I won't mangle it somehow. :) I've tried to compress this as much as possible without being confusing.

The goal of humanity is not simply to end up in Heaven, to reach some sort of place. It is not even necessarily the case that we are trying to restore ourselves to the state that Adam and Eve had -- for even they made a mistake at that time, a mistake serious enough to have cosmic ramifications.

So what was the problem? Well we do know of Adam and Eve being cast from Paradise after eating the fruit of the Tree of Knowledge of Good and Evil. But it's not the eating of the apple that begins the story. Rather, it was Adam's reaction to God. When God confronted Adam, Adam did not respond by saying, "Yes, I ate of the tree against Your warning. I'm sorry; please forgive me." He said instead, "The woman which You gave me, she made me eat it." Instead of being accountable for his action, for his own free will, Adam blames not only Eve but his Creator, essentially renouncing his free will. It was this act, that of failing to accept responsibility and instead casting down his neighbor, that led to humanity being cast out from Paradise -- or rather, casting itself out. Adam's action had effects throughout the cosmos, sending everything into disorder and brokenness and death.

So we see that the problem we have is that humans are cruel to one another and to God (and to creation, also). And this cannot be rectified in a way like having a bone fixed or some code swapped, for the problem is inextricably tied up in humanity's free will. So God chose to rectify this in a way that demonstrated not only His divinity, but _perfect humanity_. Christ becomes incarnate in the flesh and demonstrates, Himself, not only what it is to be truly human but that glory which all humanity is intended to receive. St. Athanasius describes this succinctly as: "God became man so that man might become God." This concept is termed theosis.

Theosis is realized in love of God and one's neighbor, actualized through humility. Unlike Adam blaming Eve, Christ, the new Adam, blames Himself. Christ takes up and covers the sin of the world, even on the Cross praying, "Father, forgive them, for they know not what they do." Christ accounts _only Himself_ as deserving of death, and all as worthy of eternal life. And in this perfect love, even death has no power over Him. Hades (a place of gloom where the dead were consigned; a jail in a sense -- not Hell) cannot contain Him and is destroyed, and those who were bound there by Adam's sin are released. "Christ is Risen from the dead, trampling down death by death, and upon those in the tombs bestowing life." It through these actions that all of creation was redeemed.

Yes, that was "redemption" in the past tense. From the Orthodox perspective, redemption has already occurred because Christ has not only rectified the division between God and man, He has given us the tools to experience divine life even now, through humble love of God and neighbor. Christ has given us the perfect example of how to live, He has destroyed death, and through Christ's union with humanity in the Incarnation, descent into Hades, Resurrection, and Ascention, paved the way to become by grace that which God is by nature.

"The powerlessness of Christ has always threatened the powerfulness of the world. And the silence of the cross is the most eloquent sermon about the power of love. Despite what we know in ourselves and whatever we see in our world, the cross proclaims what love can and will achieve. The scandal of the cross is that, in spite of our wrongs and the wrongdoings of our world, God loves us to the point of death." --Fr John Chryssavgis, Light Through Darkness: the Orthodox Tradition