And even in a mythical Age there must be some enigmas, as there always are. <PERSON> is one (intentionally).
-Letter 144
That said, I think we can show what <PERSON> is not.
<PERSON> is not a man, hobbit, elf, or dwarf
...or any other sort of "normal" creature.
The ring had no effect on him.
<PERSON> is not a Maia
<PERSON> was also a Maia but he was tempted by the <PERSON>. Likewise, <PERSON>, the <PERSON>, and <PERSON> were able to seduced by the Darkness.
In contrast, <PERSON> has little interest in the affairs or "politics" of Middle Earth.
<PERSON> is not a Valar
As far as it is told, the Valar live exclusively in Valinor. After the destruction of the Two Lamps, the Valar never interacted directly with the affairs of Middle Earth. And after Númenóreans reached Valinor, it was removed from the rest of Arda.
<PERSON> is not <PERSON>
<PERSON> was not all-powerful, at least according to the Rivendell elves:
"But in any case," said <PERSON>, "to send the Ring to him would only postpone the day of evil. He is far away. We could not now take it back to him, unguessed, unmarked by any spy. And even if we could, soon or late the Lord of the Rings would learn of its hiding place and would bend all his power towards it. Could that power be defied by <PERSON> alone? I think not. I think that in the end, if all else is conquered, Bombadil will fall, Last as he was First; and then Night will come."
-Fellowship of the Ring
Of course, <PERSON> could be wrong.
But <PERSON> as <PERSON> would be rather inconsistent with other manifestations of divine beings. 381
Generally, the more powerful the being, the further removed from Middle Earth he is. The wizards (Maia) could not contend with Sauron directly. The more powerful <PERSON> -- <PERSON> included -- did not live in Middle Earth at all.
Clues
Though <PERSON> and his origins may be deliberately an enigma, that doesn't mean there aren't clues painting a vague idea.
Recall that <PERSON>'s said he would be "Last as he was First".
In the same meeting, <PERSON> says,
<PERSON>, I think, is right. Power to defy our Enemy is not in him, unless such power is in the earth itself.
So the clues:
* <PERSON> is unaffected by normal "human" emotions/ambitions
* <PERSON> is very, very old, even "First"
* <PERSON> cares more about nature than politics or people
* <PERSON> is compared to "the earth itself"
<PERSON> is an incarnation of Arda, or perhaps a steward of Arda, or of Middle Earth. He's a "Mother Earth"/"Father Time" figure of sorts. We can't know exactly, since it is supposed to a mystery. But that's the gist. Human emotions have no more effect on him than they do on a tree, or rock. He is as old or nearly as old as the Earth itself, he enjoys nature, and he cares for it. And he has been there for a very, very long time.
EDIT: I found http://www.cas.unt.edu/~hargrove/bombadil.html, which describes him similarly, as a Pan-like "nature god" 247