The situation in The Lord of the Rings was pretty dire. The free people of Middle-earth had possession of Sauron's One Ring, but that was only a minor advantage. Sauron had several powerful servants, and their combined armies were vast. Many believe that the Dark Lord didn't even need the One Ring to conquer every last corner of Middle-earth.
Because outlasting Sauron wasn't an option, some people wanted to use the Ring. Boromir, for example, wanted to deliver the Ring to Gondor where it could be wielded against the Dark Lord. Gandalf and Elrond, however, knew that wasn't a realistic option. So, the Fellowship set out, and the quest succeeded with Frodo destroying the One Ring in the fires of Mount Doom. But what if Frodo had failed in the last moment and had claimed the One Ring as his own?
If Frodo Claimed the One Ring, the Nazgûl Would Have Feigned Loyalty
To be perfectly blunt, Frodo did fail. Once inside the Sammath Naur, he refused to destroy the Ring. Instead, he put it on his finger. If not for Gollum biting off Frodo's finger and Eru Ilúvatar tripping Gollum, Frodo would have declared himself the master of Middle-earth. The quest would officially have failed, and things would have ended much differently.
Tolkien actually addressed this hypothetical situation in Letter 246. The first thing to happen would have been the arrival of the Nazgûl. As soon as Frodo put the One Ring on, Sauron would have felt his presence in Mordor and sent the Nazgûl after the troublesome Hobbit. However, when they arrived, the Nazgûl couldn't have done much. Sauron unequivocally commanded their allegiance, but the Nazgûl were also tied to the One Ring.
So, Tolkien envisioned a scenario where the Nazgûl greeted Frodo as a Dark Lord. Then, they would have coaxed him away from the Sammath Naur (to ensure the Ring's survival) and feigned submission. Next, the Nazgûl would have lured him to Barad-dûr, or Sauron would have personally come to settle the score.
Sauron Verses Frodo Would Have Ended Badly
If Sauron had shown up, all bets would have been off. Frodo would have been utterly destroyed for several reasons. He was physically weak and, compared to Sauron, his will was naught. Even with the power of the Ring at his disposal, Frodo would have utterly failed because he simply didn't know how to command the One Ring's domineering power. Here's a passage from Letter 246 that shows just how brutal a Sauron versus Frodo match would have been:
"In any case a confrontation of Frodo and Sauron would soon have taken place, if the Ring was intact. Its result was inevitable. Frodo would have been utterly overthrown: crushed to dust, or preserved in torment as a gibbering slave. Sauron would not have feared the Ring! It was his own and under his will. Even from afar he had an effect upon it, to make it work for its return to himself."
Tolkien explaining Frodo's theoretical downfall shows a few things. First off, Gandalf was right to let Gollum live in Moria. Without him, this hypothetical scenario wouldn't have been hypothetical. Second, Gandalf and Galadriel were right to refuse the Ring. With its power, they might have been able to stand up to Sauron, and they might have failed. But both outcomes would have seen Middle-earth ruled by a terribly powerful figure. Thus, the only way to achieve peace was by destroying the One Ring.