The three main villains:
Kuja - Revealed at the end of disk 1. You must defeat him because, well duh, he wants to be the most powerfull and later on destroy everything.
Lifa Tree/ Necron - Revealed at the end of disk 2. It controls the FFIX's Lifestream, or something. The party tried to prevent it by destroying a little boss out there, but that solved nothing. They HAD to defeat the central function of the Lifa Tree, the mechanism that interefered with the Cycle of Souls. Aka, Necron. They finally could met the mechanism once they died, in the spiritual realm/world, which was right after Kuja killed them with an Ultima.
Garland - Revealed at the end of the disk 3, although made an appearance at +/- the beginning of that disk. Creator of Kuja, and the Lifa Tree/ Necron. He must be destroyed just alongside the Lifa Tree/ Necron if the party wants to save Gaia from Terra.