At the end of the Lord of the Rings books and movies, Sauron is defeated when the Ring is destroyed. But is he gone forever or does he survive in some form? Either way would be interesting and would fit with the bittersweet nature of the story's ending. However, only one answer can be the right one. To understand the answer, it is important to understand who Sauron really is in Lord of the Rings. It is equally crucial to understand Gandalf's purpose on Middle Earth and what it has to do with Sauron and the Ring. Sauron is not just any kind of Dark Lord. Technically, he is the Middle Earth version of a fallen angel, a Maia. He served the Vala Melkor, also known as Morgoth, who is pretty much the Middle Earth version of the Devil. After the destruction of his master at the end of the First Age, Sauron himself wanted to conquer Middle Earth.
In order to do so, he created the Rings of Power and the One Ring. He gave the Rings of Power to the Races of Middle Earth, the Elves, Dwarves, and Men, and kept the One Ring for himself. The Elves, however, figured out what he had done as soon as he put on the One Ring and therefore took off their Rings. Sauron was also unable to control the Dwarves, but he did cause them a lot of trouble, as the wearers of the Rings became greedy. However, the nine Men who got Rings ended up turning into the Nazgûl and serving Sauron. Their names are not known, except for their leader, who is called the Witch-King of Angmar. He is defeated in the end by Éowyn and Merry. Sauron's defeat takes a lot more and has many more moving parts than his servant the Witch-King's. But is this defeat permanent? Or does Sauron survive in some form?
That depends on what Lord of the Rings says about the Maiar, and it's not much. Yet other books of Tolkien's such as The Silmarillion do discuss them. In the world of Lord of the Rings, there is a creator god who is known as Eru Ilúvatar, who first created beings known as the Ainur, or the Valar and the Maiar, who helped Ilúvatar with the creation of the world. One of the Valar, however, turned evil. That was Sauron's master Melkor, also known as Morgoth. Sauron, therefore, is a Maiar, one of the Ainur who are not as powerful as the Valar but are still pretty intimidating. Sauron is one of the most powerful Maiar and learned a lot from the Vala Aulë the Smith. All of the Ainur are immortal which makes Sauron's fate interesting. The Lord of the Rings story itself might answer the question of what happens to Sauron after the Ring is destroyed. Gandalf seems to believe that with the destruction of the Ring, Sauron would become an evil spirit who would not be able to take any physical form again.
This, if it is indeed Sauron's fate, would mean that he could no longer interfere in the affairs of Middle Earth. And it might be. If anyone would know it would be Gandalf. Not only is Gandalf an intelligent being, but his entire purpose in Middle Earth is to defeat Sauron. Gandalf is actually a Maia and an Istar or wizard sent to Middle Earth to help the people of the land fight Sauron. Therefore, his entire purpose revolves around Sauron and the Ring. It is crucial to Gandalf that Sauron is defeated, which is why he is resurrected until his purpose is complete. Afterward, Gandalf does not remain on Middle Earth for very long. He crowns Aragorn, reunites with his other friends in the Fellowship of the Ring, and in the books, goes to speak to Tom Bombadil. It is unknown what he does between that time and when he meets Frodo and the other Hobbits to take the ship with Frodo to The Undying Lands.
Sauron does survive in some form after the destruction of the Ring. However, since he put so much of his own power into the Ring, he only exists as an evil spirit and cannot do anything. Instead of having control over all the people of Middle Earth, Sauron barely has control over his own fate. A fitting end for the villain of Lord of the Rings. Though he does not perish entirely, he is no longer a threat. And therefore Gandalf has completed his mission and the people of Middle Earth are safe. The fact that Sauron isn't entirely destroyed, however, shows Tolkien's philosophy when faced with evil. Do not destroy the evildoer, but take away their ability to do further harm. Ensure that they face consequences of some kind, but do not become harmful in the quest to undo the evil doer's harm. This is a good way of looking at things and could inspire many Lord of the Rings fans who deal with harmful people in their own lives.
Though most people would not be faced with Sauron-like evil, there are definitely some bad people in the real world. And any Lord of the Rings fans who come into contact with them could be inspired by the heroes' defeat but not the destruction of Sauron to do something similar. Of course, not every Lord of the Rings fan will agree about Sauron's fate. Some might believe he was destroyed entirely. While others might not see his defeat but not complete destruction as very significant or inspiring. That's fine, everyone's different. Yet given what is known about Sauron and about Gandalf's mission on Middle Earth, this explanation definitely makes a lot of sense. After all, Sauron, being a Maia, would be difficult to entirely destroy. And Gandalf, if Sauron would just reform again, would have had to stay on Middle Earth. The fact that he left confirms that Sauron is no longer a threat to anyone.