Ability materia, simply called "Ability" in-game, is an item that can be equipped to units with ability slots. This page lists not only ability materia available, but also includes abilities obtained from all equipment, other materia, and espers as well.
Magic abilities require units to have their affinity level equal or greater of the listed value to equip, while special abilities have no requirement whatsoever. This doesn't apply to equipment or espers.