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. In the episode, the statues fell and broke because Patrick bumped into them. They also appear in Battle for Bikini Bottom as containers that can be smashed. They were created for Squidward to make him happier, but after Patrick wrecked them Squidward became a lot angrier.
A Tiki Statue is made out of wood or stone. They are also colored.