Each Limit Break level from 1-3 has two Limit Breaks available. Using Cloud as an example here, his first level is Braver and Cross-Slash.
To unlock the second limit break in a level, you just use the first enough. To unlock a new level, you must have that character defeat enemies until the level unlocks. I believe this only starts counting after you've unlocked both limits on the level.
Level 4 is different. You have to find an item for each character and use it. But... not all level 4 limits are created equally. Some characters might be better off staying at level 3.
Vincent and Cait Sith are different. Vincent only has one limit per level and Cait only has two limits at all, his ultimate limit is a level 2 limit rather than a 4.
The Set command lets you set your limit level. The higher it is, the more damage you have to take to break, but you get to use the limits of that level. It's rare you want to set this lower than what you've unlocked, but there are certain times where you may want to.
