Invisibility and the Ring!

[Maédhros]

I have always wondered about the One Ring and the fact that it could made a Human or a Hobbit "invisible". How could a ring do such a thing? Would an elf be rendered invisible too?
Let us look at Sauron:
He's an ëalar
From Morgoth's Ring: Later Quentas

'spirit' (not incarnate, which was fëa, S[indarin] fae). ëala 'being'.

Sauron decided to clothe himself, he became incarnated himself, and the longer a spirit uses a hröa, it becomes more depended on it.
Because Sauron was a maia, (one of the most powerful), when he made his Ring, he had the ability, if you will, to control his spirit and hröa more than an elf or a man, and that is why he wouldn't become "invisible" when he put it on.
Let's take a look at the Children of Ilúvatar:
From Morgoth's Ring: Myths Transformed

The nature of an Elvish fëa was to endure the world to the end, and an Elvish hroa was also longeval by nature; so that an Elvish fëa finding that its hroa endured with it, supporting its indwelling and remaining unwearied in bodily delight, would have increased and more lasting joy

Men on the other hand, had another path. Their fëar were not meant to endure the lenght of Arda, so after a while, when their bodies were slained, they left Arda. Also Men had less control over their hröa than an elf would.
From Morgoth's Ring: Later Quentas

Indeed in their earlier days death came more readily; for their bodies were then less different from the bodies of Men, and the command of their spirits over their bodies less complete.
This command was, nonetheless, at all times greater than it has ever been among Men. From their beginnings the chief difference between Elves and Men lay in the fate and nature of their spirits. The fëar of the Elves were destined to dwell in Arda for all the life of Arda, and the death of the flesh did not abrogate that destiny. Their fëar were tenacious therefore of life 'in the raiment of Arda', and far excelled the spirits of Men in power over that 'raiment', even from the first days protecting their bodies from many ills and assaults (such as disease), and healing them swiftly of injuries, so that they recovered from wounds that would have proved fatal to Men.

Ok, so now we know the characteristics of the fëar and hröar on Ainur, Elves and Men. Let's see the "invisible" part of it.
From LOTR: A Knife in the Dark

Immediately, though everything else remained as before, dim and dark, the shapes became terribly clear. He was able to see beneath their black wrappings. There were five tall figures: two standing on the lip of the dell, three advancing. In their white faces burned keen and merciless eyes; under their mantles were long grey robes; upon their grey hairs were helms of silver; in their haggard hands were swords of steel. Their eyes fell on him and pierced him, as they rushed towards him. Desperate, he drew his own sword, and it seemed to him that it flickered red, as if it was a firebrand. Two of the figures halted. The third was taller than the others: his hair was long and gleaming and on his helm was a crown. In one hand he held a long sword, and in the other a knife; both the knife and the hand that held it glowed with a pale light. He sprang forward and bore down on Frodo.

Notice that when Frodo puts the Ring on, he is "transported" to the spirit world, he can see the fëar of the Nazgûl. It is my belief that because Frodo is a hobbit, he has not the control over his hröa as an elf or a maia would, and therefore he is unable to control the Ring and therefore he became "invisible". This applies to Men too because Hobbits are related to Men.
My guess is that an experienced elf or a Maia could wield the Ring without becoming "invisible".
What about the effects of the Ring on the fëar and hröar of a Men?
From Morgoth's Ring: Myths Transformed

Very soon then the fëa and hröa of a Man in Aman would not be united and at peace, but would be opposed, to the great pain of both. The hröa being in full vigour and joy of life would cling to the fëa, lest its departure should bring death; and against death it would revolt as would a great beast in full life either flee from the hunter or turn savagely upon him. But the fëa would be as it were in prison, becoming ever more weary of all the delights of the hröa, until they were loathsome to it, longing ever more and more to be gone, until even those matters for its thought that it received through the hröa and its senses became meaningless.

The ring grants you more life, but does it? The ring prolongs the endurance of the hröa but it's not a good thing for a Men or Hobbit because they become trapped by it, and in time they begin to feel the effects of it like Bilbo and Gollum.
From LOTR:Many Meetings

'I am old, Gandalf. I don’t look it, but I am beginning to feel it in my heart of hearts. Well-preserved indeed!’ he snorted. ‘Why, I feel all thin, sort of stretched, if you know what I mean: like butter that has been scraped over too much bread. That can’t be right. I need a change, or something.’

My conclusion is that the Rings grants you in a certain way, the scenario planted of what would happen if a Men was granted immortality of the hröa, (up to a certain point). Something that goes against the will of Ilúvatar ends bad. The wearer would not be blessed but cursed.
From LOTR: A Shadow of the Past

In Eregion long ago many Elven-rings were made, magic rings as you call them, and they were, of course, of various kinds: some more potent and some less. The lesser rings were only essays in the craft before it was full-grown, and to the Elven-smiths they were but trifles - yet still to my mind dangerous for mortals. But the Great Rings, the Rings of Power, they were perilous.
‘A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings. Yes, sooner or later - later, if he is strong or well-meaning to begin with, but neither strength nor good purpose will last - sooner or later the dark power will devour him.’

Looking at that quote, I would say that if an Elf would put on the One Ring, it would not become invisible, as a mortal would. As I have said before, I think it has to do with the fact that Elves had a better mastery of their hröa than a mortal would.