Sauron is the Big Bad of The Lord of the Rings. In fact, he is the Lord of the Rings. He is also known as Annatar, Aulendil, Gorthaur, Mairon, Thu, the Lord of Wolves, and a lot of other names and titles. He is first mentioned in published material in The Hobbit, where he is called 'the Necromancer'. He is an extremely powerful being who is very skilled at persuading people to do things that are bad for both themselves and the world.
Sauron is originally a Maia who serves Aulë, the Vala of craftsmanship. However, he leaves and becomes one of Melkor's chief servants. At first he acts as a spy, but he later openly serves Melkor. He is the first commander of Angband, but after Utumno is destroyed, Melkor takes it as his main fortress. Later in the Dagor Bragollach, he captures the Elves' fortress of Minas Tirith (not to be confused with the later one) and occupies it with demonic wolves.
When Beren, Finrod Felagund, and their company try to get by him, he captures them and kills all but Beren. However, he is defeated by Luthien and Huan, and only his talent at spin enables him to survive Morgoth's wrath. After Morgoth is defeated by the Valar, Sauron repents, but he is unwilling to undergo the humiliation of going back to Valinor. Instead, he hides in Middle-earth, which he regards as abandoned by the Valar.
Sauron decides to help heal the continent, but his impatience and lingering desire for domination cause him to backslide. After gathering power among the remnants of Morgoth's forces in the East, he decides to go to the Elves and pose as an emissary from Aulë under the name Annatar, Lord of Gifts. Gil-Galad, Elrond, and Galadriel mistrust him, but many of the smiths of Eregion gladly work with him. One of them is Fëanor's grandson, Celebrimbor. With them he makes sixteen of the Rings of Power, but secretly he makes the One Ring, which contains much of his power.
The Elves also make three Rings without him, the last and greatest. When Sauron puts on the One Ring, the Elves realize his true intentions and hide the Rings. This causes Sauron to invade Eriador, capturing Celebrimbor and torturing him in an attempt to find out where the Rings, particularly the last three, are; Celebrimbor tells him where the other Rings, which Sauron had already helped make and tainted, are, but refuses to tell him where the Three are. Angered, Sauron kills him and uses his body as a flag when he presses on towards Lindon. Thankfully, the Númenorean Men help the Elves drive him away, but Eregion is destroyed.
Later, the Númenoreans, led by their king Ar-Pharazon, return to destroy him. Sauron's armies desert, and he surrenders to Ar-Pharazon and convinces the king to bring him back to Númenor as a prisoner. Before too long, he goes from being a prisoner to being Ar-Pharazon's most trusted adviser and chief priest in a Morgoth-worshiping cult. Then he convinces Ar-Pharazon to attack the Valar, and Eru Ilúvatar responds by destroying Númenor and making Arda round.
Sauron re-establishes himself in Mordor and fights against the Last Alliance between the Elves and the few living, non-Morgothist Númenoreans. Sauron kills Gil-galad and Elendil, but is defeated by Isildur, who, not knowing the Ring's precise nature, takes it as a weregild for his father.
Greatly weakened, Sauron moves to what becomes Dol Guldur, a fortress near Mirkwood. There he works to recover some of his former power and becomes known as the Necromancer. The Elves and Wizards know that something evil lives there, but suspect – or hope – it is a Nazgûl. Meanwhile, Sauron sends the actual Ringwraiths to attack Arnor and Gondor. These manage to capture Minas Ithil, thus winning Sauron a palantir.
Later, Gandalf manages to work his way into Dol Guldur, realizes it is Sauron, and reports back to the Elves and other Wizards. They – after Saruman stalls them for a while – attack Dol Guldur, and Sauron leaves for Mordor. When Saruman acquires the land of Isengard, Sauron manages to use the palantir there to convince Saruman to join his side, though the Wizard still plans to take the Ring himself. Meanwhile, Sauron rebuilds his armies.
From Gollum he learns that the One Ring is held by 'Baggins' from the 'Shire', and he sends the Nazgûl to find the Ring. Sauron believes that, logically, his enemies will unite under one strong leader who uses the Ring. From his point of view, no Ringbearer would be able to make themselves destroy it, even if for some reason they decided to do so. Aragorn is the obvious choice of Ringbearer, so when the united armies of Gondor and Rohan march on the Black Gate of Mordor, Sauron figures that only someone who had the Ring would do something so bold.
This distracts him, allowing Frodo and Sam to destroy the Ring. This causes him to be destroyed, reducing him to a 'mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape'.
On the Eye of Sauron
In the books, Sauron has a humanoid shape, except when he chooses to shift it. During the events of The Lord of the Rings, he looks like a terrible dark lord with four fingers on one hand, having lost the ability to shift into a fair form. His Eye serves as a symbol of his power, attention, and will, not an actual physical form (though this is debated).
In the movie, the metaphor is, for whatever reason, made into a giant, flaming, disembodied Eye atop Barad-dûr. This is apparently his physical(ish) form in the movieverse.
Many Mary Sues claim to be his daughter. Other Mary Sues reform him with the power of Love or turn evil. Sometimes he's even woobiefied. This is despite the fact that, as mentioned above, he is a giant, flaming, disembodied Eye in the movieverse (which is likely the only Lord of the Rings universe such Sues are familiar with).