We know that for hobbits, the One Ring makes them invisible. But, at least according to the movie, Sauron was not invisible when he was wearing the ring. What effects would the ring have on other types of people: Dwarves, Elves, magic users, or any other interesting category one might have.
When Frodo and Gandalf were discussing the Ring in Fellowship of the Ring, Gandalf said that all of the Rings of Power granted power proportional to the ability of the wearer. A simple proto-hobbit like Smeagol got invisibility and little else; Frodo got a little more initially, probably due to his experience with the Elves, and gained more abilities as his experience expanded (whereas Samwise, on his one use of it, got only invisibility just as Smeagol did). Someone like Gandalf, a Maia on a par with Sauron, would have control over his visibility; in fact, we know he did, because in the end it was revealed that he wore one of the Elven-rings (likewise Galadriel who also wore one).
In the book, Tom Bombadil (A creature of uncertain origin, but high power,) Wears the ring, and does not disappear. Frodo says that "He has power over the ring," but Gandalf corrects him, saying rather, "the ring has no power over him."
Dwarves are stated to be resistant to being changed from outside, and thus do not become invisible.
Having not read the Simarillion or any other Tolkien works outside of the trilogy & The Hobbit, I believe the effects of the ring are similar no matter who wears it.
The Ring is invested with great power and many abilities. It can do many things, though for most people it only extends their lives and turns them invisible. It is also intelligent to some degree, possessing "a will of its own". The Ring wants to return to Sauron.
My speculative theory is that the ring acts upon its user in whatever way it thinks will make it valuable. It uses this to move around, always to places where it can be seen by more people or seized by the right people.
The Ring influences Isildur to take it away from the battle and not destroy it. So it turns him invisible (showing him it can be hidden with ease, while bestowing great power).
With Gollum, it gives invisiblitiy as well - he's on the run and paranoid about people stealing from him, invisibility is invaluable.
With Bilbo, it grants invisibility because he needs to sneak around.
Frodo KNOWS it's a ring of invisibility, and it gives him what he expects.
Those who wear it and have great power don't turn invisible - they can prevent it from overtly affecting them, and presumably it offers them its other powers instead (or in addition).
It seems to tune what it gives to its wearer depending on the wearer (Frodo's shadow never showed up while he was invisible, but Bilbo didn't get a big Ringwraith beacon on his finger when he wore it), and it seems to choose invisibility in most cases.