Short answers, pulled from the story.
The Nazgûl, also called the Ringwraiths or the Nine Riders, are nine men who accepted Rings of Power from the Dark Lord Sauron. The rings extended their lives but enslaved their wills and reduced them to invisible wraiths bound to the One Ring. The Lord of the Rings calls them Sauron's "most terrible servants."
The leader of the Nazgûl is the Witch-king of Angmar, also called the Lord of the Nazgûl and the Black Captain. He served as Sauron's chief agent for most of the Third Age, destroying the successor kingdoms of Arnor and directing the assault on Minas Tirith. Tolkien speculated, in notes for translators, that the Witch-king was of Númenórean origin.
The Witch-king is destroyed at the Battle of the Pelennor Fields by Éowyn of Rohan and the hobbit Merry Brandybuck. Merry strikes him with an enchanted Númenórean blade, and Éowyn drives her sword between his crown and mantle. This fulfills the prophecy of the Elf-lord Glorfindel, who had declared that the Witch-king would not fall "by the hand of man."
Scholar Edward Pettit, writing in Mallorn, identifies the Lacnunga, an Old English book of spells Tolkien knew, as a source for several features of the Nazgûl. The "Nine Herbs Charm" from the Lacnunga invokes nine as the key mystic number in Germanic lore, and the charm "Against a Sudden Stabbing Pain" contributed the image of thundering riders, the figure of the Witch-king, and the logic behind the Morgul-knife.
Nazgûl is drawn from the Black Speech of Mordor, combining nazg, meaning ring, and gûl, meaning wraith. Scholar Jason Fisher traces "wraith" through Old English, Gothic, and Old Saxon roots all meaning crooked, bent, or twisted, and connects the English homophone "ghoul" to the Arabic ḡūl, a demon that feeds on corpses.
The flying mounts of the Nazgûl have no formal name in Tolkien's text, where they appear only as "fell beasts." The soldier Beregond of Gondor called them "Hell Hawks," and in a letter Tolkien referred to them as "Nazgûl-birds." Tolkien described the creatures as naked and featherless with hide-webbed wings, and acknowledged they were "obviously pterodactylic," while speculating they might be "a last survivor of older geological eras."