First, a bit of background: Tom Bombadil is the oldest being in Arda, and is probably the most powerful. Tom Bombadil once said to Frodo, "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." suggesting that he was already an inhabitant Arda when Yavanna made the first growing things (trees, flowers, bushes, etc). He also has unparalleled control over his domain, exceeding that of Sauron, and even Melian and Thingol of old. He met Frodo and company when they were ensnared by a powerful and malevolent Huorn, who Tom calls "Old Man Willow". He has many different names, and possibly a different personality to go with it. The Elves call him Iarwain Ben-adar, which literally means, "old person without a father"(Iarwain means old-young, Ben means without, and adar means father). The Dwarves called him Forn, which means "Eldest", and the Men called him Orald, which means "Elder". Now, knowing this, there are several theories that say who Tom Bombadil is, and what his origin story is.
Tom is a Valar
One theory is that Tom is a Valar, specifically Aulë. I'm guessing that why people think that is because Tom is so powerful he must be a Valar. And Goldberry must be Yavanna, because she likes nature, and might even be able to influence it. But Aulë is the Smith, the Lord of Metalworking, while Tom Bombadil seems to be more into nature than weaponry. In fact, we never even see him touch metal, and the house which he lives in might not even have metal in it. And, Goldberry can't be Yavanna, because Goldberry is the "River-Daughter", while Yavanna was created directly by Eru Ilúvatar. As for the other Valar, no other Valar have the right skill set and interests as Tom. The closest we come to seeing one is with Oromë, but Oromë is the Great Rider, while we never see Tom riding on a horse, only Fatty Lumpkin, his pony, whom he seldom rides anyway. If he was Oromë, I think we would see a lot more hose action. Not to mention that all the Valar are in Valanor.
Tom is a Maiar
Ah yes, a theory that actually holds some water. Some think that Tom Bombadil is just a Maiar that doesn't live in Valinor. But I don't think that is true, because even Sauron, who was originally one of the most powerful Maia (Mairon), was under the influence of the One Ring, while the Ring had no grip on Tom's mind. Also, even while Sauron and Melian had very good control over their lands (Melian's Girdle, and Sauron literally lives in a realm of fire where the sun never shines), none of them can even rival the way that Tom is in his kingdom. At least this theory has some form of logic to it, unlike the last one. All of the Maiar are likely in Valinor. Also, if he were a Maia, he would be more influential in Arda, because the other Maiar who were in Arda were extremely influential (Sauron and Melian).
Tom is Father Nature
This theory isn't half-bad. The theory says that Tom is like an embodiment of Arda, kind of like a father nature. It makes sense, too. Tom is the master of his domain, and Galdor once said, 'Power to defy our enemy is not in him, unless such power is in the earth itself. And yet we see that Sauron can torture and destroy the very hills.' suggesting that Tom is connected to the earth. Every part of it fits perfectly into this notion. The only reason it isn't my favorite is because of how sound this next theory is.
Tom the embodiment of the Music of the Ainur
This theory is my favorite. The theory states that Tom Bombadil is the music of the Ainur, put into a body. The theory works perfectly. It would explain why he has such a huge music aptitude, as the Music is inside of him, woven into his very being. It would also describe how the beasts and beings of the Old Forest obey him, as "his songs are stronger songs," a direct quote from Tom himself. There are also linguistic clues that say this. When Frodo asks Goldberry who Tom is, she replies saying, "He is", in the common speech. But if she had been speaking Elvish, she would have said 'Eä', which is what the universe of Tolkien's legendarium is called. And, if Tom was made from the Music of the Ainur, then, as Ungoliant is a lot like the anti-Tom, it would make sense that Ungoliant would be made from the Discord of Melkor (Morgoth).
One last note: Tom Bombadil does not have his own origin. Tolkien did not make one for him. Tolkien wrote in one of his letters, “even in a mythical Age there must be some enigmas, as there always are. Tom Bombadil is one of them (intentionally)."