Final Fantasy X has a Condition Turn-based (CTB)-battle system, by which only the time exceeds when orders during the battle are carried out. The order of the turns from the allies and the enemies are shown in the CTB-display at the top right of the screen.
The order changes often, depending what the ally or enemy does. For example if you carry out a fierce, powerful attackit take time to restore before the next action. You can cast Haste on your characters to get more turns during battle.
You can also change party members during battle by using the L1 button. It's very handy for if you want to gain AP with all members or when your pary member is weak or isn't good enough. It's never been this way in previous Final Fantasy's.
Also you can change your equipment during battle which is a nice addition if you have the wrong equipment equipped.