Tom Bombadil is, as Tolkein said, an enigma.
But he is obviously VERY important and powerful to Arda.
People have denied it, and exactly because they deny it I affirm it, but I believe Bombadil to be none other than Eru Illuvatar Himself.
He is so ordinary, so happy, so...out of place in the World and yet so endeared to it...I can conclude nothing else.
He is, after all Eldest of All, and was there to see the first raindrop, even there before Morgoth entered the world.
People say Tolkein denied it. Yes, he claimed no embodiment for God in his Arda. And yet...Tolkein himself often changed his mind and contradicted himself and made changes to his mythology. Also, Tolkein implied (though he waffled on this) that even Tolkein himself was not entirely sure what Tom was. An enigma. Perhaps Tolkein did subconsciously put a manifestation of God, merry and simple and utterly transcendent of all other categories, into his World. Not necessarily an "incarnation" (Tolkein would have insisted on one (Christian) incarnation of Jesus Christ at a later age even in his fictional mythology) but a manifestation or an unexplained existence or symbol of Eru Himself.
And it doesn't really matter what Tolkein said, by the way. Post-modern criticism of literature will tell you that once the author has written the text, they lose control over its "Truth". It then becomes an interaction with the reader (as vast fandoms indicate) and to me, the most meaningful interpretation, the most beautiful and sublime, is that Tom is Eru Illuvatur himself.
He is not affected by the Ring. He enjoys his life in his little corner, and lets everything else run its course, helping Good but mainly letting them figure it out for themselves.
At the Council they will not entrust the ring to him, and that makes sense...if he is God, even they might not know. And if they do, it still might not make sense if the ring must be Destroyed. They say that if he held the ring, there would be no place for him in the world anymore as Sauron would destroy all and then Tom "Last as he was First"...But this could perhaps be taken as a meaning that God cannot simply "hold" evil...it must be fought and destroyed in his Creatures...or else if all goodness in his creation is destroyed, God himself would cease to be the Good Creator, and would have lost his battle. Impossible? Perhaps, the point is such a loss was destined NOT to happen as good did triumph.
What is Tom? He is.
To me, he must be a literary manifestation of God. Uncertain, imprecise, and not fitting into the world. The Mystery of Mysteries, the Sign of Ultimate Contradiction. A kindly old man who lives in a forest singing ridiculous songs.