Short answers, pulled from the story.
Tom Bombadil first appeared in "The Adventures of Tom Bombadil", a poem published in The Oxford Magazine in 1934. His wife Goldberry, Old Man Willow, and the barrow-wight were all present in that original poem, years before The Lord of the Rings was written.
Gandalf explains in The Lord of the Rings that "the Ring has no power over him". The Tolkien scholar Verlyn Flieger argues this is because Bombadil has no desire to dominate anything. W. Christopher Stewart adds that Bombadil pursues knowledge purely for its own sake and has no interest in power, which is the Ring's only purpose.
Jackson explained that he and his co-writers felt Bombadil did little to advance the story, and including him would make the films unnecessarily long. Christopher Lee agreed, noting the scenes were left out to make time for showing Saruman's capture of Gandalf.
The folklore scholar David Elton Gay argued that Bombadil was directly modelled on Vainamoinen, the demigod at the center of Elias Lonnrot's 1849 Kalevala. Both are the oldest immortal beings in their worlds, exercising power through song and knowledge. Each lives in a small forested territory he controls but does not own.
Tom Bombadil is a supporting character in The Fellowship of the Ring who lives in the Old Forest with his wife Goldberry. He rescues Frodo's company from Old Man Willow and a barrow-wight and hosts them for two nights. The One Ring has no power over him, a fact that prompts debate at the Council of Elrond.
Robert Foster suggested in 1978 that Bombadil is one of the Maiar, the angelic beings sent from Valinor. Gene Hargrove argued in Mythlore in 1986 that he is one of the Valar, specifically Aule, the archangelic demigod who created the dwarves. Tolkien himself said only that Bombadil is an intentional enigma who does not fit any of the categories he created.