What is Tom Bombadil?

He's just a supremely powerful being (nameless thing, perhaps) who was created at the same time as Arda and who is just content living in a forest singing all day about how hot his wife is instead of caring about anything that happens in the world around him.

The question is, what is his wife, Goldberry? She appears to be a personification of nature, Arda, or just the Old Forest or something.