In The Silmarillion, "Of the Rings of Power and the Third Age," we see this description of Isildur's desire to keep the Ring:

The Ruling Ring passed out of the knowledge even of the Wise in that age; yet it was not unmade. For Isildur would not surrender it to Elrond and Círdan who stood by. They counselled him to cast it into the fire of Orodruin nigh at hand, in which it had been forged, so that it should perish, and the power of Sauron be for ever diminished, and he should remain only as a shadow of malice in the wilderness. But Isildur refused this counsel, saying: ‘This I will have as were-gild for my father's death, and my brothers. Was it not I that dealt the Enemy his death-blow?' And the Ring that he held seemed to him exceedingly fair to look on; and he would not suffer it to be destroyed.

From this, we can see that Isildur seemed either largely ignorant of the true nature of the Ring, or he simply disbelieved that Sauron could return while the Ring was whole. It's power immediately entranced him, and he became enamored of it. He refused to yield what he saw as the "spoils of battle."

In Fellowship of the Ring, "The Council of Elrond," we are told of what Isildur himself wrote regarding his feelings for the Ring:

And after these words Isildur described the Ring, such as he found it.

"It was hot when I first took it, hot as a glede, and my hand was scorched, so that I doubt if ever again I shall be free of the pain of it. Yet even as I write it is cooled, and it seemeth to shrink, though it loseth neither its beauty nor its shape. Already the writing upon it, which at first was as clear as red flame, fadeth and is now only barely to be read. It is fashioned in an elven-script of Eregion, for they have no letters in Mordor for such subtle work; but the language is unknown to me. I deem it to be a tongue of the Black Land, since it is foul and uncouth. What evil it saith I do not know; but I trace here a copy of it, lest it fade beyond recall. The Ring misseth, maybe, the heat of Sauron's hand, which was black and yet burned like fire, and so Gil-galad was destroyed; and maybe were the gold made hot again, the writing would be refreshed. But for my part I will risk no hurt to this thing: of all the works of Sauron the only fair. It is precious to me, though I buy it with great pain."

Isildur was entranced by the Ring; by its beauty, by its mystical nature. Again, he seems to be fully ignorant of the Ring's nature.

We are given a glimpse into the nature of the Ring's power from Gandalf in FotR, "The Shadow of the Past." Gandalf explains Gollum's relationship with the Ring:

'All the "great secrets" under the mountains had turned out to be just empty night: there was nothing more to find out, nothing worth doing, only nasty furtive eating and resentful remembering. He was altogether wretched. He hated the dark, and he hated light more: he hated everything, and the Ring most of all.'

'What do you mean?' said Frodo. 'Surely the Ring was his precious and the only thing he cared for? But if he hated it, why didn't he get rid of it, or go away and leave it?'

'You ought to begin to understand, Frodo, after all you have heard,' said Gandalf. 'He hated it and loved it, as he hated and loved himself. He could not get rid of it. He had no will left in the matter. 'A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it. At most he plays with the idea of handing it on to someone else's care – and that only at an early stage, when it first begins to grip. But as far as I know Bilbo alone in history has ever gone beyond playing, and really done it. He needed all my help, too. And even so he would never have just forsaken it, or cast it aside. It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him.'

We must consider, the Ring is not just an ordinary piece of jewelry; it contains a portion of Sauron's power as a Maiar. Part of Sauron's power is the ability to dominate others wills. By creating the Rings of Power, he meant to instill each Ring with part of his will, and tie them all to the Ruling Ring, the One. Merely possessing the Ring was an invitation to that will of Sauron to control you. It was such a subtle control, that you would scarcely be aware of it. You would find it beautiful, or precious, and be unable to see the evil of it. Sauron was a great deceiver throughout his narrative; the Ring also possessed this aspect of its creator. Frodo was resilient to this power in two ways: he was a humble and good spirited Hobbit, and also he was aware of the Rings corrupting nature from the start of his possession. He willed himself to resist its siren song, whenever he could. Even with a strong, pure will, he gave in occasionally. By the time he arrived at Mt. Doom, his will was exhausted with struggling with Sauron's, and he gave in fully, although because of the desire of Gollum for the Ring, Frodo lost it, and it was destroyed. The prophecy of Gandalf about the Ring, that it "could not be willingly destroyed," proved true: only through a greed-fueled accident was the Ring finally undone.