Cloth can be worn by every class in the game, however, only 3 classes are restricted to Cloth: Mages, Priests and Warlocks. While every other class can wear Cloth, there are no benefits to wearing it.
Leather can be worn by every class apart from Mages, Priests, and Warlocks. It is the primary armor of Rogues and Druids, while Hunters and Shamans use it primarily until Level 40, when they learn Mail.
Mail can only be worn by Death Knights, Paladins, and Warriors, with Hunters and Shamans able to use it from Level 40. Paladins and Warriors use it primarily until Level 40, while Death Knights will practically have no use for Mail, wearing Plate throughout their game journey.