The mechanics of this is never made clear; suffice to say that the Ring is a magic ring, and the invisibility is an area-of-effect spell.
It also generally doesn't vary from character to character, although there are a handful of exceptions:
Isildur himself becomes invisible but one of the items on his person, the Elendilmir, does not:
Isildur turned west, and drawing up the Ring that hung in a wallet from a fine chain about his neck, he set it upon his finger with a cry of pain, and was never seen again by any eye upon Middle-earth. But the Elendilmir of the West could not be quenched, and suddenly it blazed forth red and wrathful as a burning star.
Unfinished Tales Part 3: The Third Age Chapter 1: "Disaster of the Gladden Fields"
Why the Elendilmir cannot be quenched isn't explained, but we know that it's an Elvish-made crystal; presumably there's some magical property that reacts poorly to evil magic rings.
Bilbo, much like Isildur, was turned invisible by the Ring. His sword Sting, on the other hand, was not; it became visible when he unsheathed it:
With that [Bilbo] turned and found that the last space between two tall trees had been closed with a web—but luckily not a proper web, only great strands of double-thick spider-rope run hastily backwards and forwards from trunk to trunk. Out came his little sword. He slashed the threads to pieces and went off singing.
The spiders saw the sword, though I don’t suppose they knew what it was, and at once the whole lot of them came hurrying after the hobbit along the ground and the branches, hairy legs waving, nippers and spinners snapping, eyes popping, full of froth and rage.
The Hobbit Chapter 8: "Flies and Spiders"
Since Sting is an Elvish blade, forged in Gondolin, the reason for this exception is presumably similar to the reason for the Elendilmir.
Tom Bombadil, notably, is completely unaffected by the Ring:
Then Tom put the Ring round the end of his little finger and held it up to the candlelight. For a moment the hobbits noticed nothing strange about this. Then they gasped. There was no sign of Tom disappearing!
Fellowship of the Ring Book I Chapter 7:"In the House of Tom Bombadil"
Although much of Bombadil's nature is a mystery, Gandalf gives us a possible explanation for this exception later on:
'It seems that [Bombadil] has a power even over the Ring.'
'No, I should not put it so,' said Gandalf. 'Say rather that the Ring has no power over him. He is his own master.
Fellowship of the Ring Book II Chapter 2: "The Council of Elrond"
Sauron himself doesn't become invisible. This is discussed in a related question
However, in the general case, it clearly does. Although this full idea didn't survive to the final version, in an early draft (the earliest draft, in fact), Tolkien suggests that turning the wearer's clothes invisible really is an effect of the ring; the ringwraiths, who have become naturally invisible due to prolonged use of a ring of power1, still require the use of a ring to turn their clothes invisible:
Yes, if the Ring overcomes you, you yourself become permanently invisible - and it is a horrible cold feeling. Everything becomes very faint like grey ghost pictures against the black background in which you live; but you can smell more clearly than you can hear or see. You have no power however like a Ring of making other things invisible: you are a ringwraith; and your clothes are visible, unless the Lord lends you a ring. But you are under the command of the Lord of the Rings.
History of Middle-earth VI The Return of the Shadow Chapter III: "Of Gollum and the Ring"
1 In this draft the distinction between the One Ring and the other rings hadn't been established; all of these rings are effectively the same, and Bilbo's is just the last (intact) one that Sauron hasn't recovered.