Fellbeast

In J. R. R. Tolkien's The Lord of the Rings, the term fell beast is used to describe the gigantic flying pterosaur-like creatures on which the Nazgûl rode on after being unhorsed at the Ford of Bruinen. It is especially prominent during the Battle of the Pelennor Fields, where the Witch-king of Angmar, the Lord of the Nazgûl, rides his in battle against King Théoden of Rohan.

In book V, chapter 6 "The Battle of the Pelennor Fields" in The Lord of the Rings, Tolkien describes it thus:
 * "[...] it was a winged creature: if bird, then greater than all other birds, and it was naked, and neither quill nor feather did it bear, and its vast pinions were as webs of hide between horned fingers; and it stank. A creature of an older world maybe it was [...]"

The fell beast attacked and eventually killed Snowmane, Théoden's horse. The horse fell on Théoden, crushing him. Dernhelm (who revealed herself as Éowyn) defended the dying Théoden, and challenged and killed the beast.

Earlier, in The Fellowship of the Ring, at the River Anduin, Legolas shoots one down in the night as it approaches them.

Tolkien actually only used the name "fell beast" as a descriptor and not an actual name. ("Fell" in this sense is an archaic English word meaning "cruel", "evil" or "lethal".) However, since this creature lacked any real name, "fell beast" is often used to refer to it.

It is never explicitly stated that all the Nazgûl rode fell pterosaur-like creatures, but, of course, one can infer that they did so.

1978 cartoon
In Ralph Bakshi's 1978 animated version of The Lord of the Rings, one of the Nazgûl (possibly the Witch-king, for he carries a mace), is shown riding a fell beast. However, Bakshi's film only covers events up to the Battle of the Hornburg, so that is the last we see of the fell beasts and their riders.

1980 TV special
In the Rankin-Bass 1980 animated version of The Return of the King, the Nazgûl ride winged horses.

Live-action movies
In Peter Jackson's trilogy of movies based on The Lord of the Rings, the fell beasts are depicted as more dragon-like and serpentine creatures, i.e. their heads appear more like a snake's and they don't have beaks, leading to the common misconception that they are dragons in the books; Tolkien definitely meant a pterosaur-like creature, as shown in the quote above.

Also, they are much larger than is implied in the books, where they are essentially used as light observation planes. In the movies, they are used for attack much more often, with the Nazgûl usually swooping down and screeching, disorienting the defenders of Minas Tirith. In the books, the main weapon of the Nazgûl is fear, not sonic waves, and fear is what makes the defenders at the gate of Minas Tirith run away, leaving Gandalf alone to face the Witch-king as he enters.

Although on screen the films never make this mistake, sometimes actors on the commentary tracks refer to the fell beast as a Nazgûl; this is incorrect. The fell beast is the creature that the nine Nazgûl ride, and the mistake probably arose because fell beasts are always seen with a Nazgûl atop them.

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