The actions of each character (that you control) are determined by the Judgement Ring. This powerful symbol reflects one's chances of performing a task, whether it be as calculated as an attack or as simple as opening a gate. Either way, when it comes to use the Judgement Ring, timing is everything.
When it comes time to use the Judgement Ring, sections of the ring will be highlighted. The object of using the Judgement Ring is to stop the cursor in the highlighted areas by pressing the X button. As the meter spins around the ring, press the X button as many times as there are highlighted areas to complete your action. The closer you can get the meter to the outside edge (the red part) of the highlighted area, the better your action will be. This is very difficult to do, but will be extremely useful in a fight. Also, if you are in a fight make sure you check the meter in the first area, as this will give you atleast one attack. If you don't check the first area, you will not be able to launch an attack.
The ring is different for each character and task, and it changes a little each time you switch weapons on your character. Make sure you know where to check the meter based upon the task you are trying to accomplish. It's difficult, but with a little practice you'll have no trouble.