My theory on the invisibility factor of the one ring when worn by the hobbits and Gollum, is that the intense power of the one ring swallowed up the one who wore it. For example, no one but Sauron was powerful enough to withstand being swallowed up by it.
Terrijayne's explaination is almost correct in a way. There is a parallel plane which lies alongside Middle Earth, referred by the likes of Gandalf as the Wraith world. The One Ring transports its wearer into this world, and hence they become "invisible" because they have entered this shadow world. This is why Frodo was able to see the Nazgul on Weathertop when he put the ring on. He had entered their world.
It is not simply a case either of power that stops the Ring turning its wearer invisible. This is an assumption people make because neither Sauron or Tom Bombadil were turned invisible by it. Both Sauron and Tom are Maiar spirits. They are naturally invisible, because in their true form they are spirits who live in the "Wraith" world. Maiar, however, are able to weave themselves an image that can be seen by the people of Middle earth because it is projected into the "Real" world. What the Hobbits see when they meet Tom is not really Tom, but a projected facade. When he puts the ring on, therefore, he does not appear to turn invisible because he is already in the "wraith" world. His facade is something that is being projected out from that world and is not effected therefore.
Galadriel is of the "real" world, so if she wore the One Ring, she would turn invisible. She is a powerful sorceress, however, so she might have sufficient power to project a facade out of this shadow world as Tom and Sauron do. Gandalf and Saruman are Maiar too, but they have been given the burden of flesh while they are in Middle Earth. This flesh is real, not a facade (although I believe in the case of Gandalf the White, it perhaps is), so they too would be effected by the invisibility property.
As for the Elven Rings, they do not have this property on the wearer, but do appear invisible themselves to those who are not aware of them. They do, however, grant the wearer and those around them the power of invisibility from scrying. That is how both Galadriel and Elrond were able to keep their respective realms hidden from Sauron's prying eye for so long, and why Sauron was unable to find Gandalf even though he entered Dol Guldur on a couple of occasions.