Who or what is Tom Bombadil?

Tom Bombadil is certainly not a hobbit, although as with many characters created by Tolkien, his origins are uncertain. Elrond says, "I had forgotten Bombadil, if indeed this is the same being who walked the woods and hills long ago, and even then was older than the old. That was not then his name. Iarwain Ben-adar we called him, oldest and fatherless."

Elrond was born some 50 years or so before the end of the First Age, and at the time of LOTR only Cirdan, Galadriel and Treebeard, and of course Bombadil, are older (I discount the Wizards here). For a person who is about to approach his 6,500th birthday, (why are there no elf candle-makers?) to refer to another who is called 'eldest and fatherless' implies a very great span of time indeed.

As the only real contact elves had with hobbits at the time of LOTR (aside from Bilbo) was to pass through their lands on the way to the Gray Havens, and considering the relationship elves in LOTR have with the hobbit characters (rather paternal/maternal, kindly but a bit patronising) this idea of him being a hobbit doesn't quite fit. Tom himself claimed to have been alive since the very early First Age, and, like Treebeard, remembered a time when the whole of Middle Earth was covered in forest. Hobbits lived naturally (without magical assistance) to about 100, so that doesn't fit either.

No definite answer is provided by Tolkien but clues are there. Tom is absolute master of his own realm - The Old Forest - and in the chapter 'The Council of Elrond' it is suggested by Erestor that he should perhaps be given the ring to safeguard as -

"It seems he has a power even over the Ring."
"No, I should not put it so," said Gandalf. "Say rather that the Ring has no power over him. He is his own master."

No-one in LOTR (with this exception) is immune from the power of the ring. Elrond and Galadriel refuse it, even Gandalf is tempted by it. Boromir sucumbs to it's lure. Bilbo is still suffering from his possession of it, even 20 years after giving it up, and we know what happened to poor Smaegol, who possessed it (or did it possess him?) for about 500 years.

Robert Foster suggests (and I think it quite likely) that, given his undoubted great age and the respect with which he was viewed by such characters as Elrond, Galadriel and Gandalf, Bombadil was most likely to have been a Maia who, enchanted with the natural life of Middle Earth, had abandoned his brethren when they went to live in Aman, (The Undying Lands) after the destruction of the Two Lamps. If this is the case, he would have seen the western migration of the first elves, and with the possible exception of Treebeard, would be the oldest physical creature in Middle Earth at the time of LOTR.

I think Tom Bombadil is Aule and Goldberry is Yavanna

Tom Bombadil is the oldest living creature on middle earth Tom was the First and Tom will be the last.
He watched all living things arrive in middle earth.

http://www.cas.unt.edu/~hargrove/bombadil.html

Tom Bombadil is a hobbit ...i think...read chapters 6 and 7 from the first book