This is a very hard topic to talk about.
From the Encyclopaedia of Arda
1. Bombadil Within Tolkien's Cosmology
Tolkien's universe is inhabited by a multitude of races and beings: our problem is that what we know of Tom does not fit easily with any of these. He seems almost to have been 'transplanted' from elsewhere. In fact, this is almost certainly what happened, at least in a literary sense, but at this point we are concerned primarily with giving Tom a place within Tolkien's universe.
Though there are many candidates to choose from, we can at least dismiss most of these immediately. Tom is definitely not a Man, a Hobbit, a Dwarf, or indeed of any mortal kind, and we can also take it for granted, for obvious reasons, that he is not an Orc, a Troll, an Ent, a Dragon or an Eagle! But this still leaves plenty of possibilities:
Was Tom an Elf?
Tom's capering, his wisdom, his great age and his love of song undoubtedly give him a certainly 'Elvish' quality. This possibility though, is easily disproved by the following from The Lord of the Rings:
"'When the Elves passed westward, Tom was here already...'"
Tom's own words, from The Fellowship of the Ring I 7, In the House of Tom Bombadil
Tom would hardly have said this if he was an Elf himself! This is, incidentally, proof of Tom's great age - the Elves 'passed westward' in the Great Journey some six Ages before he spoke these words.
Was Tom a Maia?
This a very common suggestion, to the extent that it is sometimes treated almost as 'fact'. There is, though, no direct evidence for this - it seems to be based on the idea that since Tom can't be a Vala, and there is no other possibility, he must be a Maia. As we'll see, these are both flawed assumptions - Tom might be a Vala, and there is at least one other possibility.
Though we can't say for certain that Tom wasn't one of the Maiar, there are grave difficulties with this position. The most important of these is that the Ring had no effect on him:
"Then Tom put the Ring round the end of his little finger and held it up to the candlelight... There was no sign of Tom disappearing!"
The Fellowship of the Ring I 7, In the House of Tom Bombadil
There were other mighty Maiar in Middle-earth at the time of the War of the Ring, especially Sauron, Saruman and Gandalf, and all of these were in some sense under the power of the Ring. Yet Tom is unaffected by its power of invisibility, nor does he feel any desire to keep it (he hands it back to Frodo 'with a smile'). Tolkien himself points out the importance of Tom's immunity. On this topic, he says:
"The power of the Ring over all concerned, even the Wizards or Emissaries, is not a delusion - but it is not the whole picture, even of the then state and content of that part of the Universe."
The Letters of J.R.R. Tolkien, No 153, dated 1954
Was Tom a Vala?
The last of Tolkien's named races (using the term loosely) that might include Tom is that of the Valar, the Powers of the World. A common argument against this is that we know the names of all the Valar, and Tom isn't among them. This doesn't hold water:
"...[the Valar] have other names in the speech of the Elves in Middle-earth, and their names among Men are manifold."
The Silmarillion, Valaquenta
While of Tom himself it is said:
"'[Bombadil] was not then his name. Iarwain Ben-adar we called him, oldest and fatherless. But many another name he has since been given by other folk...'"
Elrond, from The Fellowship of the Ring II 2, The Council of Elrond
It isn't inconceivable, then, that Tom is one of the fourteen known Valar, dwelling incognito in Middle-earth. Though we can't be certain, it seems likely that a Vala would be capable of resisting the power of the Ring, and so that difficulty can be set aside. The 'Vala Hypothesis', though, is not without difficulties of its own, with perhaps the most significant being:
"'Eldest, that's what I am... Tom remembers the first raindrop and the first acorn... He knew the dark under the stars when it was fearless - before the Dark Lord came from Outside.'"
The Fellowship of the Ring I 7, In the House of Tom Bombadil
All of the beings who became Valar existed before Arda was made, so any of them could with justification claim the title 'Eldest'. But Tom says he 'knew the dark under the stars' (that is, he was in the World, not outside it) 'before the Dark Lord came from Outside'. The term 'Dark Lord' is uncertain here - it might apply to either Melkor or Sauron, and both originally came from 'Outside' the World. If he means Melkor, then this is very significant: consider this description of the entry of the Valar into the World, from the original conception of the Silmarillion:
"Now swiftly as they fared, Melko was there before them..."
The Book of Lost Tales, Part I, III The Coming of the Valar and the Building of Valinor
'They' here refers to Manwë and Varda, who were explicitly the first Valar to enter Arda apart from Melko (Melkor). In Tolkien's original conception, then (and there is nothing in the published Silmarillion to contradict this) Melkor was the first being from 'Outside' to enter the World, and yet Tom suggests that he was already here when Melkor arrived!
Admittedly Tom may be referring to Sauron, who must have come to Arda after these great ones, but the phrase 'before the Dark Lord came from Outside' seems to make more sense if he means Melkor (that is, he is referring to an event of cosmic significance, and a specific point in the World's history, which isn't the case with Sauron).
This is only one of the objections to the Vala theory. Another, for example, is that characters who we would expect to recognize a Vala living in their midst (especially Gandalf) don't apparently do so.
There are many other arguments to be made both for and against Tom's status as a Vala. For a more detailed discussion of this topic, and some more concrete conclusions, Eugene Hargrove's fascinating essay Who is Tom Bombadil? is strongly recommended.
Was Tom Ilúvatar Himself?
Tom's powers are apparently limitless, at least within his own domain, and this has led a lot of people of suggest that he might be none other than Eru Ilúvatar himself. There are certainly several hints in the text of The Lord of the Rings that this might be the case; he is called 'Master', and 'Eldest', and Goldberry says of him simply;
"'He is.'"
The Lord of the Rings I 7, In the House of Tom Bombadil
All of these points might suggest that Tom and Ilúvatar were in some sense the same being. In fact, though, this is one of the very few theories about Tom that we can bring to a definite conclusion. This point is touched on several times in Tolkien's letters, and each time he makes it clear that Tom and Eru should not be confused. Perhaps his most definite statement is this:
"There is no embodiment of the One, of God, who indeed remains remote, outside the World, and only directly accessible to the Valar or Rulers."
The Letters of J.R.R. Tolkien No 181, dated 1956
If there is no embodiment of the One (that is, Eru), then Tom cannot of course be such an embodiment.
Was Tom a 'Spirit'?
The idea that Tom might be a 'spirit' (as opposed to a Maia or Vala) is certainly possible according to The Silmarillion. Though it seems to be commonly assumed that only the Valar and the Maiar entered Arda, a tantalising glimpse of Tolkien's original vision survived into the published form of the work. Here, discussing the Aratar or eight mightiest Valar, he says:
"...in majesty they are peers, surpassing beyond compare all others, whether of the Valar and the Maiar, or of any other order that Ilúvatar has sent into Eä."
The Silmarillion, Valaquenta
This single phrase 'any other order' seems to be a survival of a much older and more detailed account found in the Lost Tales:
"...brownies, fays, pixies, leprawns, and what else are they not called, for their number is very great... they were born before the world and are older than its oldest, and are not of it, but laugh at it much..."
The Book of Lost Tales, Part I, III The Coming of the Valar and the Building of Valinor
It is hard not to hear the echo of Tom Bombadil in these words, and perhaps here we see the first germ of his inspiration (the Lost Tales predate Tom's first appearance in print by about a decade). Whether Tom is a brownie, fay, pixie or leprawn, though, is open to doubt - none of these creatures appears in Tolkien's published works, and their function as a bridge to later folklore seems to have been taken up, at least partly, by the Hobbits.
This version of the 'spirit' idea doesn't address many of the other problems already discussed, though. Why should a 'leprawn' be immune to the Ring when the Maiar are not? Could a 'brownie' have entered the World before the first of the Valar?
There is another kind of spirit that Tom could be though: a 'spirit of nature'. Tolkien himself seems to support this point of view:
"Do you think Tom Bombadil, the spirit of the (vanishing) Oxford and Berkshire countryside, could be made into the hero of a story?"
The Letters of J.R.R. Tolkien, No 19, dated 1937
This letter predates Tom's appearance in The Lord of the Rings (in fact, this quotation is part of discussion of the possible sequel to The Hobbit), so it is at best circumstantial evidence.
The idea of a 'nature spirit', though, is certainly possible within Tolkien's universe. Though this area of his cosmology is never directly addressed, Middle-earth seems at times to be full of spirits - at least some trees apparently have spirits, for example (consider Old Man Willow, or the Huorns of Fangorn). Consider too:
"'But the Elves of this land were of a race strange to us of the silvan folk, and the trees and the grass do not now remember them. Only I hear the stones lament them..."
Legolas, from The Fellowship of the Ring II 3, The Ring Goes South
There are numerous other examples of this kind: it is clear that in Tolkien's universe, the stuff of nature is somehow more alive, and more aware, than in the modern world. It is a short step from this to the idea of 'spirits of nature', but a much longer one to 'spirits of nature' that wear yellow boots and live in houses.
Gosh that's deep.