Costumes are skins for your monsters. They don't boost a monsters stats in any way, however, and are only for appearance. Buying Cardboard Boxes, Toy Boxes, Washing Machines, and Top Secret Boxes are currently the only ways to acquire costumes.
There are three rarities of costumes:
Special costumes are only obtainable through getting all common, rare, or legendary costumes of one monster.
Washing Machine boxes will guarantee at least one rare costume. Top Secret Boxes will always award you with three legendary costumes.