Everyone do yourself a favor and don't even try to watch any of the previous versions of The Hobbit and/or Lord of the Rings movies. They're utterly awful. Ralph Bakshi should be locked in a room somewhere and forced to watch that abomination until his end.
Regarding Tom Bombadil, it's hard to say who/what he is. Let's review what we know: Bombadil is very, very, very old. He's lived in that general area of the world for ages and ages, near as anyone can tell. That's an important clue, because in the Second Age (the events of The Hobbit & LOTR occur in the Third Age) Sauron was even more powerful than he was during LOTR. He had a huge army, overran all of Eriador (where The Shire is, basically everything west of the Misty Mountains) and overthrew everything there (an Elvish kingdom), and laid siege to both the Grey Havens and Rivendell. Despite this near utter control of the region, there is zero mention of any capture of Bombadil or any flight by him away. He just seems to have remained where he was and nobody bothered him.
Jump-cut to "modern day" and Frodo and Co. go bumbling about the Old Forest, get captured by Old Man Willow, but then quite easily rescued by Tom Bombadil - who rescues them by singing a song - and then invites them over for supper. The song is also a clue, since we're told that in the very beginning of all things, creation was made by a song - the song of Illuvatar and the Eldar. Songs are powerful, and the powerful use songs.
So we're at Bombadil's house, rescued from the Willow, and the topic of the Ring comes up. Frodo tries to trick Tom by putting the Ring on and sneaking out of the room, but Tom sees him despite the Ring's power and then demonstrates that the Ring has no power over him with some basic parlor tricks.
To understand all this, we must first understand several other things. First, what is the Ring? The Ring is, simply put, the manifestation of the greater part of Sauron's power. What is Sauron? Saurons is a Maia, one of the "angels," if you will, of the first creation, second only in power to the Valar. The Valar are, of course, the Lords of Arda (creation), demigods ruling over Arda for Illuvatar, The One, the greatest being who began all things. So that puts Sauron in context, but it's also important to note that, among Maiar, Sauron was the most powerful. Gandalf and Saruman were also Maiar, but far less powerful than Sauron. The Balrog was also a Maia, but Gandalf overthrew him in single combat, so that puts some context in the order of Maiar. Gandalf was a badass, but Sauron would have taken him in the fourth round, probably by knockout. No TKO; he'd just put him down.
So the Ring is the manifestation of the greater part of the might of one of the most mighty beings on Middle-Earth, and as such a tremendously powerful artifact. Yet it has no power over Bombadil. Why is this? There are several possible answers. One deals with Tom's description of himself as "the oldest." Tom says that he was there before anyone else, anything else. Combine that knowledge with Tom's use of song as power, and his complete indifference to the Ring and its power, and you have very few conclusions available to you. Namely:
1. Tom is one of the Valar. This is a common theory, but as with all of these theories, it is not supported by fact. The Ring would have no power over a Vala. Valar are known to use songs as power. Tom is not only stronger than the creatures (sometimes terrible creatures) around him, but completely unafraid of them and unaffected by them. One of the Valar theories says that Tom is actually Aule, the Vala of "making," closely associated with the world and nature. In that context, Goldberry would be Yavanna, another Vala wholly connected with growing things. This theory has merit, but the problem with it is Galdor's statement in the Council of Elrond that (paraphrasing) "The power to withstand [Sauron] is not in him." If Bombadil were Aule he could most certainly withstand Sauron - in fact, Aule was at one time Sauron's chief, before Sauron "went to the dark side" with Morgoth. Aule would crush Sauron like a bug. So Bombadil is not Aule. Further, we're told that once the Lamps were destroyed in the First Age (read The Silmarillion), the Valar didn't come to Middle-Earth much at all - yet Tom lives there, and has lived there for Ages. So the likelihood that Tom is a Vala is pretty slim.
2. Bombadil is Illuvatar. This is pretty easily dismissed, as Illuvatar, the Greatest of Gods, The One, the Grand High Poobah of All, never entered his creation. This theory has less legs than the Vala theory.
3. Bombadil is Arda itself. Or, more clearly, Bombadil is a manifestation of Arda, a spirit born of the very existence of Arda. This has some merit thanks to Tom's assertion that he was there "first," but there's pretty much no other evidence to support this.
There are several more theories. I'd jot down more, but I'm running out of time, so I'll wrap up with this - in his copious notes and letters, Tolkien discloses several things about Bombadil. He says that Bombadil (as with many other things in his stories) was first a plaything of his children, a toy that he wanted to include in his story to amuse his kids. So that's the mundane beginning of the Bombadil idea, but it doesn't tell us much.
And that's where every search ends. With "not much." And that's intentional, at least on the part of Tolkien, because he stated on more than one occasion that every story should have some questions that are left unanswered. He believed strongly in myth, and that the maintenance of "things unknown" was important. Tolkien himself claims to have known who/what Bombadil is, but he also took that knowledge to the grave - or, at least, anyone he's told (like Christopher, perhaps), isn't telling.
And I doubt we'll ever truly know.
