Final Fantasy VI
Final Fantasy VI was the first game to have some form of "Limit Break" system. However, in their original incarnation they were referred to as Desperation Attacks.
Unlike versions that have surfaced since, Desperation Attacks were quite simplistic in their execution. To perform a Desperation Attack, a character had to appear in weakened state. The most common reason for this was due to have low amounts of HP and if this status was achieved, the game would at random choose to perform a Desperation Attack instead of a normal attack.
As the named suggests, these attacks therefore came in times of desperation and the damage was greatly amplified compared to a normal attack.
Not Everyone Can Perform Desperation Attacks
Although Desperate Attacks were available for most of the characters, not all of them had access to such moves. Both Gau and Umaro were without Desperate Attacks, but everyone else had completely unique attacks which could cause substantial damage to enemies.
The only thing that really differs about each of the moves aside from their names is the length of the animation. They are for the most part quite short, but some are longer than others.