Final Fantasy VI
Although the term "Limit Break" comes from Final Fantasy VII, a similar concept had already appeared in Final Fantasy VI, where it was known as Desperation Attacks. All character's that have the Attack command have the chance to do an extremely critical attack when their health is critically low and the Attack command is used. Unlike later Final Fantasy games, the player only has a 1-in-16 chance of performing a Desperation Attack when in critical status, and not every character has one.
Final Fantasy VII
The first game to coin the actual phrase "Limit Break" and define the concept for the rest of the series. In Final Fantasy VII, each character has a Limit Gauge that fills up when the party member takes damage. When the gauge is filled a special ability will become available for the character to use. The first time the gauge fills up, the ATB meter for that character rapidly fills so they are the first to select a command. Also the Limit Break takes priority over any other command that may have been input before the Limit was selected. This game also introduced multiple Limit Breaks per character and a separate leveling system. To earn new Limit Breaks, the character had to either use the current Limit Break a set amount of times or physically kill 80 enemies. Level 4 Breaks are unlocked via special items, typically at the end of a side quest.
Final Fantasy VIII
In Final Fantasy VIII the Limit Gauge was removed. For the most part, when a character's heath is very low, and mostly in the yellow "critical" state a limit will become available by hitting the right arrow instead of choosing a basic attack. There are a some random times when health is fine that a Limit will become available, but the player must be careful because if they accidentally hit attack it will not be there the next time around. Different Limit Breaks could be unlocked by improving weapons.
Final Fantasy IX
In Final Fantasy IX's Limit Breaks are now called Trance Mode. It is similar to Final Fantasy VII's style in that it has a gauge that fills up, but Instead of using a single ability, during Trance a character's appearance changes and they have a purple glow to them. They will be in this state for a few turns and have access to special abilities. Trance takes a role in the story of the game.
Final Fantasy X
Continuing with the gauge that fills up, Final Fantasy X's Limits are known as Overdrive. But it brings it back to a single ability that the character can preform.
Final Fantasy XII
Limit Breaks in Final Fantasy XII are known as Quickenings. To perform them, characters must unlock them on their license boards and are tied to Mist Charges, related to the MP bar. Level 2 Quickenings, for example, take 2 Mist Charges or approximately 50% of the character's MP. Quickenings were designed to introduce real-time inputs into the limit break animations, similar to some of the inputs in Final Fantasy VIII and Final Fantasy X. This system also found its way into Final Fantasy XII: Revenant Wings.