A Tiki Statue just appears in the episode: Enchanted Tiki Dreams. They are all on the Enchanted Tiki Island. Squidward's House is also a Tiki Statue, though not as detailed. They fell and broke because Patrick bumped into them. They also appear in Battle for Bikini Bottom as containers that can be smashed.
A Tiki Statue is made out of wood or stone. They are also colored.