If a player moves through multiple squares threatened by a single opponent, does that player provoke in each square he moves through or only the first? I became confused when I saw the wording in the core rules that discusses how Combat Reflexes interacts with this.
A character can usually only make one AoO per round, or more with Combat Reflexes or other abilities. But that's not the real question - what provokes an AoO?
An opponent can provoke multiple AoOs from the same person in a given round easily, see the Pathfinder SRD section for the relevant rules. Generally you only provoke one AoO from a given opponent per ACTION (not per round). So if you are in melee with someone, and guzzle a potion and then run away, you provoke twice. One move action only provokes once, even though you may move out of multiple threatened squares. Heck, if you are getting extra actions somehow, you can provoke some more AoOs too. Whether they get to take all those AoOs is dependent on them, but provoking them is dependent on your actions.
May I quote the SRD, emphasis mine:
But if you have Combat Reflexes, and someone provokes two AoOs from you in a round (which is trivially done if they are goons), you get two extra attacks on them. In fact there are other feats and abilities that can cause movement that provokes AoOs not during the opponent's turn - if that happens you can get a whole raft of AoOs in the same round on them. (For example, Greater Bull Rush.)
Also note you don't have to take an AoO if you don't want to - you can ignore someone who provokes and save your AoO for later (like if a bunch of people are charging through your line and you want to stab the second guy in line instead of the first one).
Normally a character may make only one attack of opportunity per round - period. Combat Reflexes allows the PC to make one additional attack of opportunity per round up to the character's Dexterity mod. (1) (2)