A caster is someone who casts spells. Warlocks, Priests, and Mages are traditionally considered the "caster classes." Additionally, Druids, Shamans, and Paladins have spells and might also be considered casters, although they also have the potential to play as a melee-based character.
Caster use Mana to fuel their spells, and cannot cast spells while silenced. While Hunters also use mana, they are not considered casters because their abilities mostly deal physical damage, have no cast time, and are not affected by silence effects.
Caster can also be used to described a mob who primarily casts spells.