Backstory and Motivation
Zemus appears only briefly during the entirety of Final Fantasy IV. As a Lunarian, he inhabits an artificially constructed moon circling the game's mythos' version of Earth after his homeworld destroyed. He and other Lunarians created the moon in question as a temporary resting place as the race continued to look for a new place to call home. While some of his fellow Lunarians paid a visit to Earth and wished to co-exist with its people, introducing them to new technologies, Zemus wished to claim Earth for strictly Lunarian use--eradicating humanity in the process.
Mythos in Final Fantasy IV (Potential story spoilers)
Zemus makes his only appearance at the very end of Final Fantasy IV, leading up to the final battle. Prior to this, when Cecil's party first makes it to the moon and encounters the elder Lunarian sage, FuSoYa, it is revealed that Zemus had been forced into a slumber in order to prevent his taking over Earth. Meanwhile, crystals on the moon worked together to bind his powers. This kept Zemus trapped deep within the moon near its core, which is ostensibly why he never makes a direct appearance during most of the game. His effects were felt throughout the game, due to his still being able to exert telepathic control over more vulnerable characters--primarily Golbez--despite having his powers bound. His initial plan was to summon the Giant of Babil, a massively destructive machine capable of decimating Earth, thereby clearing out the planet for Zemus and his kin to populate it. After the failure of this plan, FuSoYa and a recovered Golbez take to the moon to confront Zemus, and seemingly defeat him just as Cecil's party arrives to watch the supposed victory.