How powerful Tom's songs are depends very much on who and what Tom is.
We don't have a definitive answer on what Tom Bombadil is. We know that he is not Eru himself. There is very powerful textual evidence that he is neither a Valar nor a Maiar. We also know that he is not an Ent, dwarf, elf, man, hobbit, orc, dragon, or any other creature we know of. So we can only resort to conjecture on just how powerful Tom's songs are.
Let us start off by assuming that Tom is indeed none of these things. Where did he come from, then?
By Tom's own words, he was in Middle-Earth "before the river and the trees." He "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 the Outside." (In The House of Tom Bombadil) So here have have evidence that Tom was in Middle Earth before the Ainur ever entered the world to shape it, especially since by some accounts Melkor was the first of the Ainur to enter the world.
There is one interesting point in the Silmarillion. Eru uses the Flame Imperishable to create independent life. Melkor is unable to get his hands on the Flame because ultimately "Iluvatar gave to their vision Being, and set it amid the Void, and the Secret Fire was sent to burn at the heart of the World; and it was called Ea." So before the Ainur entered the world, Eru set this life-giving fire at the heart of the world.
There is some discussion of Tom at the Council of Elrond. We learn that Tom has not always kept to the Old Frest; Gandalf says "now he is withdrawn to a little land, within the bounds that he has set, though none can see them, waiting perhaps for a change of days, and he will not step beyond them." So here, whatever Tom is and whatever his power is, he is not bound to the Old Forest, but we might assume he chose it for a reason. (Most likely to some degree because of Goldberry, we learn in poems.)
More interesting is what Glorfindel (who has special dispensation from the Valar to be there) says of what would happen if they gave Tom the Ring: "Could [Sauron's] power be defied by Bombadil alone? I think not. I think that in the end, if all else is conquered, Bombadil will fall, Last as he was First; and then Night will come." Galdor adds to that and says "such power 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."
So let's take a step back. Whatever Bombadil is, even people "in the know" seem to agree he was First. They also credit enormous power to Tom: he will not fall until all is conquered - he will be the Last, as he was the First, and after his fall, Night will come. The power to resist Sauron is not in him unless it is in the earth itself, but Sauron can torture the hills themselves.
One reasonable interpretation many people have taken is that Tom is an incarnation of the World itself, given life by the Imperishable Flame when Eru placed it in the World. This seems to be supported by the comments at the Council of Elrond. This explanation also has the benefit of possibly explaining the existence of other characters: Goldberry, for example, may very well be literally the daughter of the river, which might explain why Tom must gather lillies apparently to sustain her. We also know that Tolkien had in mind the spirit of the English countryside when he wrote Bombadil.
Well, then. We come to the question of what exactly such a spirit might be able to accomplish. Unfortunately, we don't really know. Tom's power seems absolute. Goldberry says of Tom that the land does not belong to him; everything belongs to itself, but "Tom is the Master of wood, water, and hill [...] nobody has ever caught old Tom [...] He has no fear. Tom Bombadil is master." An enterprising reader might notice that, though he is master of wood, water, and hill, he still exhibits power of things he is not explicitly master of: the Hobbits, the Wight, etc.
Here's what we've seen him do, from the text of the Lord of the Rings and the poems about Bombadil:
- He can command various creatures, including trees, as when he puts the Old Man Willow to sleep.
- He can travel supernaturally fast when he wishes (even he if was nearby when Frodo called, he arrived extremely quickly.)
- He knows when you sing his song. A normal person couldn't possibly have heard him.
- The Ring has no power over him - but neither has he power of the Ring's effect on others.
- He can clearly see people in the spirit world, as when Frodo put on the Ring.
- He can expel undead spirits (probably of men, but perhaps elves) to the Void. ("Lost and forgotten be, darker than the darkness, where gates stand for ever shut, till the world is mended.")
The real limitations of Tom's songs are left to our imagination - I don't think he could be said to have "power" anymore than nature itself has "power." More importantly, though, is what Tom wouldn't do - and that's most anything. :)