Well you have to have Squall in your party since he is the main character
. Um...I'll describe their Limit Breaks.
Squall's is Renzokuken (something like that). Press the trigger button when the blue bars pass. If you get them all, he will use a special attack, depending on his current weapon equipped (Rough Divide, Lion Heart. etc)
Rinoa has 2 Limit Breaks. One is using Angelo (Her dog). Angelo has a number of powers he can use, that may damage the enemy, make party invincible. etc.
Her second one is Angel Wing (Something like that). You get it in the middle of disc 3. You can't control her but she randomly casts high level spells.
Zell's is Duel. A list comes up at the bottom of the screen, showing key combos which you have to press in the right order to trigger his moves and hit the enemy. His Limit Break lasts for a certain time (10 secs I think but it freezes when he you have entered the combo)
Example:
Punch Rush: O X
Booya: Right, Left
Doing one will use that attack and stop the timer. When that attack is being shown, another list will come up showing different attacks, with timer counting down. If the combo you enter is a flashing one, the Limit Break will end, regardless of the time.
Quistis's Limit Break is Blue Magic. Basically it's Kimarhi's overdrive from FFX, learning enemy skills.
Irvine's is Shot. You need to have Ammo in your inventory to perform it. Basically you press the trigger button quickly to shoot at the enemy. You can perform different attacks depending on your ammo.
Finally,
Selphie's is Slot. A small box comes up with a spell in it. If you don't like what you get, just choose the next one. In some cases, the spells will even be cast more than once. The catch is the battle still runs so you need to be quick. She also has special spells depending on her weapon (I think)
FullCure - Restores the party's HP to full
Wall - Casts Shell and Protect on all party members.
Rapture - Puts wings on the enemy and removes them from battle fleeing.
THE END - The screen turns into the green field (Like what Rinoa is standing on in the intro movie) and the words "THE END" appear- and you won the fight. Works on Bosses
Well...usually people have Zell in there party because of his Limit Break. Then the next most used would be Irvine. I'm doing one at the moment with Quistis and Rinoa.