As we all know when Frodo slips on the one ring he disappears from sight, but why does Frodo become invisible?
The One Ring
The answer is quite simple, when the wearer puts on the ring they enter the wraith world and are hidden in the physical world.
Some beings exist in only the physical world such as humans, hobbits, many elves etc whereas others exist in just the shadow realm, such as the wraiths. Then there are others still who occupy both. Divine beings such as Sauron exist in both worlds, interestingly some of the elves also exist in both as does Gandalf. So if Gandalf was to wear the ring he would not become invisible.
We know this from the books, from passages such as this:
“You were in gravest peril while you wore the Ring, for then you were half in the wraith-world yourself.”
So why were Frodo’s clothes invisible then?
So the ring transfers the wearer partially into the shadow realm, why would this make their clothes invisible too? Well, its never really explained. The ring was originally created by Tolkein as a plot device to give Bilbo the ability to turn invisible. It would therefore also need to make any clothing invisible to work well as an invisibility device.
What would happen if Frodo never took the ring off?
Well to start with he would be in immense danger, being visible to Sauron. But if he somehow managed to evade Sauron he would slowly submit to the power of the ring, eventually becoming a wraith himself.