According to soteriologies (e.g. Calvinism) that believe God makes the choice to save people without expecting any decision from them first, what is the role of faith? If the decision making process is not mans' in the first place, why is it that God says people will only be saved by faith? What does their faith accomplish?
Put another way, why does he require obedience if there is no choice in the matter anyway?
There are a few facets to this, but the Calvinist perspective (though not nec. only the Calvinist perspective) would say that faith is the mechanism for salvation, because it gives God the glory; it doesn't give man room to boast (also, it doesn't help that man doesn't quite cut it with his works).
The apostle Paul addresses this in his letter to the Ephesians:
Ephesians 2:1-10 (ESV) [Emphasis mine]
1 And you were dead in the trespasses and sins 2 in which you once walked, following the course of this world, following the prince of the power of the air, the spirit that is now at work in the sons of disobedience— 3 among whom we all once lived in the passions of our flesh, carrying out the desires of the body and the mind, and were by nature children of wrath, like the rest of mankind. 4 But God, being rich in mercy, because of the great love with which he loved us, 5 even when we were dead in our trespasses, made us alive together with Christ—by grace you have been saved— 6 and raised us up with him and seated us with him in the heavenly places in Christ Jesus, 7 so that in the coming ages he might show the immeasurable riches of his grace in kindness toward us in Christ Jesus. 8 For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, 9 not a result of works, so that no one may boast. 10 For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand, that we should walk in them.
Reformed Theology understands the order of salvation as beginning with regeneration (being "born again" -- see John 3) with a capacity for faith. This faith is not something that is effected through the will of sinful man (who is "dead in [his]...sins"), but rather, it is a gift from God.
I think one thing that might cause confusion here is that at first glance, "faith" seems like a rather arbitrary requirement...especially since the nature of belief is such a tricky one: We only believe that which we consider to be believable (chicken or egg?), so until we find the gospel believable...why would we ever believe the gospel? It's kind of a sticky wicket from a volitional standpoint...and some different theologies seek to address this with their different takes.
Maybe more fundamentally, though, I think we can consider that God requires obedience. He created the universe, and He sets the laws of this cosmos. To violate the laws He set forth is a pretty big deal, whether we realize it or not, and so God is well within in His right to punish all disobedience (which is rebellion against the sovereign God's created order committed by the very creatures he gave life to) in whatever manner He sees fit. The Biblical account shows, though, that man is not very good at being obedient.
So, Jesus enters the scene, and provides a gracious provision. Jesus IS perfectly obedient, and he alone, fulfils the law. He also acts as the sacrificial lamb who pays the penalty for the disobedience of his people.
This is where faith comes in to play, as this defines who are Jesus's people. We are basically given a choice...do we accept the perfect life, the atoning death, and the resurection of Jesus Christ, and trust him to be our representative when judgement comes (i.e. do we stand before God as though Jesus's life record is our own -- this requires faith, as we're trusting in the validity of the testimony of Jesus, and we're trusting in him as our Saviour ), or do we reject that and choose to be our own representive -- do we rely on our own merit as the basis for our worthiness to be judged by God?
