Tom Bombadil wasn't an elf: Frodo and Strider/Aragorn both know enough about elves at the point Tom Bombadil is introduced to recognize them on sight, and they both act and behave as though he was something foreign and strange.

Tom Bombadil was not a wizard (a member of the Istari): there were only five sent to protect and guide the races of Middle Earth during the Third Age: Saruman the White, Gandalf the Grey, Radagast the Brown, and the two wizards Alatar and Pallando.

However, Tom Bombadil displays several curious characteristics that can provide some guidance as to what he is or if he fits into the rest of the Middle Earth mythology:

Everyone that seems to know him believes him to have been around for a really long time, possibly for multiple ages.

He calls the Old Forest, an offshoot of the forests that once covered Middle Earth and were the home to the Ents, his home and seems to be its master.

Additionally, everyone that seems to know him believes him to be more powerful than any other being they know (and the people that know him also know a bunch of really powerful beings)

He is completely unaffected by the One Ring (whereas really powerful beings like Sauron, Gandalf, and Saruman are)

Likewise, he seems to have very little concern for the politics of the world around him.

One theory is that he's part of the so-called Ainur: direct representatives of Ilúvatar (the creator). Depending on how you look at it, Ainur are either gods and demigods or something similar to angels; but pantheism, where the Ainur are merely different aspects of Ilúvatar, more closely describes the mythology of Middle Earth.

The Ainur are broken into two groups:

The Valar, which consist of the chief agents of Ilúvatar (think the Roman or Greek pantheon), including Lórien, Mandos, Manwë, and Morgoth/Melkor among others.

The Maiar, which consist of servants and aides of the Valar (think demi-gods or archangels). The Maiar include figures like Sauron (Morgoth's right hand) and the Istari.

There's speculation that he's a Maia, as he appears to be immortal, isn't a named Vala, and could feasibly be the agent of Yavanna (Vala of nature). However, other Maiar we're exposed to seem to be fully capable of succumbing to the One Ring's power in a way that Tom Bombadil isn't.

Other theories include that he's an incarnation of a Vala. However, it's not clear the Valar take incarnations: throughout Tolkien's works, Valar, when they want to interact, send agents or interact directly as themselves.

In reality, Tolkien never defines what Tom Bombadil is: not in The Lord of the Rings or in The Adventures of Tom Bombadil. The closest thing to a definition that Tolkien gives of Tom is that he's extremely old, and simply is.

To this end, the third and final theory, that Tom Bombadil is an incarnation of Ilúvatar himself, takes hold. Ilúvatar would definitely not be subject to the corruption of the One Ring, would be far more powerful and older than even the most powerful of the Ainur, and care little for the day-to-day struggles of the mortal and immortal races. Finally, the characteristic that Tom Bombadill simply is is similar to how Yahweh, the god of Abraham and Moses, characterizes himself: I am.

But it's best not to think of it too deeply. Tolkien was loath to create allegories and allusions, saying of The Lord of the Rings: