As a Christian who believes the Bible, I've had to try to derive the implications of several of its sayings to make cohesive theologies concerning various topics. With prayer, the Bible does have (of course) several sayings about what one should use it for and what it does. While many users above have pointed out that prayer can help the individual change himself or act as praise and worship, the Bible does seem to imply that requests can be granted, and, furthermore, if they hadn't been asked, they wouldn't have been granted.
The Bible also claims an omniscient. omnipotent, and omnibenevolent God, which, for me, returns the subject of prayer to the problem posed in the OP: if God knows everything and can do everything, wouldn't He know beforehand about the requests I would make and decide whether or not to do them anyway without my asking?
The theological system I've come up with to accommodate both God's attributes and a prayer that can result in changes relies on Bible verses that say that the God in the Bible wants His followers to ask for good things but will not give or do them unless asked. That seems imply that this God gives some measure of autonomy or even authority to those who can pray. If so, then that would allow for God to retain those attributes above stated and for prayer to affect a change at the same time.
That was dreadfully ineloquent; ask for clarification if needed.