There are many things which point to Bombadil being a mighty spirit of some kind, over whom even the ring has no power.
The only reason that the ring has power over the Istari, was that they were "incarnate", and so bound to Arda, and it's pains. Bombadil too is "incarnate" (being quite "real", and not in spirit form), yet the ring has no power over him!
In History of Middle-Earth, it is recorded that Bombadil was originally intended to be conveyed as a far relation to (Farmer) Maggot. Although this was dropped, it suggests that Tolkien did not have a high lineage planned for him. Yet after this notion of kinship was dropped, Tolkien did not explicitly state his ancestry. A point to note, is that there in only one Bombadil, and he is unique in the sense that he is of his own "race". He was certainly not Valar, since he has no power over the ring, and also he will fall "las, as he was first". No Valar will fall before the pitiful "might" of Sauron! Bombadil does not appear to be Maia either, for he was first, yet did not participate in the Silmarillion points to him not under the command of the Valar. Surely, one so "powerful" (in his own way) would have been commanded by Manwé to participate in the battle to destry Morgoth. Also, no incarnate Maia would be so free from the cares of the world. The enigma exists, I think, because he was first. How could he be first, and eldest? He was there before the first Elf saw the stars, and far before the ents. Thus he is not in the "old lists". So... Bombadil is not Valar, no Maia. Yet he is a spirit incarnate of great power, and very old, and unique. And he is not Eru. (Tolkien stating explicitly that Eru never entered into the story). He is certainly not a creature of Middle Earth. Why would Eru create a unique being, if he had no real function in Middle Earth, neither to populate it, nor to rule it? Everything seems to point to Bombadil being the spirit of Arda. The essence of the Earth. See quotes in some of the other postings. This is not without it's problems. Being the spirit of Earth, why no power over ring?? Ring is incarnate power, shown in it's goal of the domination of Middle Earth!
If Bombadil was Earth, then he would certainly care a great deal about Sauron, the orcs, and so forth. Care more than even the elves. In the end, I seems to contradict myself even more, and still arrive at no satisfactory conclusion!
Bombadil simply is.

I personally believe him to be Middle Earth incarnate, the spirit of Arda. If Goldberry is the river's spirit , then why can't Bombadil be the spirit of Earth? This then points to either a maia, and/or a valar. I make a bold statement that all such "spirits" are ainur. e.g össe == water "god" (of course under the command of Ulmo). So, cannot Bombadil be the incarnate of Earth? An Ainur, maybe of less might than Sauron. So he is a Maia, of less stature than Sauron, and is the spirit of the Earth.

Right, having "proven" the Bombadil can be neither Maia, nor spirit of the earth, I now put forward that he is both!

This only goes to prove the brilliance of Tolkien!