There is not a whole lot of information about Tom Bombadil that could give us a definitive clue as to what kind of creature he really is. We know he has a feather in his hat, a blue jacket, yellow boots, his cheeks are ruddy, his hair messy and brown, he is filled with joy and sings (what seems to be a) series of nonsense words, he has power over Old Man Willow, he can go out in the rain and not get wet, his “pretty lady” is the river woman’s daughter, the Elves call him “oldest and fatherless,” and the ring has no power over him.
But even with all of that information, we still do not have the clearest idea of who or what he is. Even Tolkien was unsure as to who or what Tom Bombadil might have been; the character having sprung up out of the ground fully formed (as it were).
There are a couple of possibilities as to what kind of creature he might be that seem more likely than any of the others.
The first is that he is some kind of nature spirit. If this were the case, it would take into account that he can command Old Man Willow to let the hobbits go, and he is obeyed. It might also be why he can be out in the rain and not get wet (but that might also be because he is in a serious relationship with the river woman’s daughter. Maybe Goldberry is the one that is keeping him dry in those cases). But if he were a nature spirit, it would explain why, when Frodo asked who he was, Goldberry said that he was the master; and the way she explained it, it sounded as though it was his responsibility to watch over the Old Forest and take care of the creatures and plant-life that resided there.
The other possibility that seems like it might be the most viable is that he may have been a Maia. Tolkien was clear that there were no angels in his Legendarium, but if you had to compare the Maia to anything in our world, the easiest way for people to understand what they are, is to compare them with one of the lesser choirs of angels. They are not so high and lofty as the Valar, but they are still creatures with more power than any in Middle-Earth, and they can survive as purely spiritual beings. Sauron, the main bad guy of Lord of the Rings, as well as the five wizards that were sent to Middle-Earth to stop him were all originally Maiar, so it is possible for Maiar to survive within Middle-Earth (and not just in places like Valinor or the Legendarium equivalent of Heaven).
The wizards were sent into Middle-Earth to help the Free Peoples fight Sauron, but Sauron came into Middle-Earth after he started following Melkor (the original Dark Lord, and one of the Valar). So, it is possible for one of the Maiar to come into Middle-Earth of their own free will; there is no requirement for them to be sent by someone in authority. If that is the case, Tom Bombadil would have snuck into Middle-Earth from somewhere like Valinor while no one was paying attention. Once there, he could have found a home that he liked and decided that he was going to stay there and watch over the place, making sure that everything ran smoothly.
But it is also entirely possible that Tom Bombadil is something completely different than either of these things, something that will continue to remain unclear forever.
What kind of creature do you think Tom Bombadil really was? Do you think he was a Maia or a nature spirit, or do you think he may have been something else entirely?