Fellbeast

The term Fell beast is used to describe the gigantic flying pterosaur-like creatures on which the Witch-King of Angmar and the other Nazgûl rode on after being unhorsed at the Ford of Bruinen in Middle-Earth.

Appearance
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 [...]"

Overview
The Fell beasts were fed on foul meats by Sauron and in The Fellowship of the Ring, at the River Anduin, Legolas shoots one fell beast down with an arrow in the night as it approaches them.

The fell beasts are 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 while the other Nazgûl fly around Minas Tirith, preying upon terrified soldiers.

After Snowmane, Theoden's horse, was killed by a black dart, the Fell Beast stooped on it's corpse, ready to devour Theoden. Dernhelm (who revealed herself as Éowyn) defended the dying Théoden, and challenged and killed the beast.

BFME III: Dawn of the Dragon
It showed the Fellbeasts orginated from the DragonRealm, as "Wyverns" a foul dragon beast without the ability to Breath fire, but has poisonous Breath. Wyverns and Fellbeasts alike lack Upper Arms (But some people think the Wings are arms) and thought are cousins or Same family tree. While Wyverns bear a poisonus barb on the end of their tails, while Fellbeasts do not...

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. But when Éowyn confronts the Witch-King he is riding a bird-like steed.

Live-action movies
In Peter Jackson's trilogy of movies based on The Lord of the Rings, the fell beasts are depicted as more Wyverns (dragon-like and serpentine creatures that walk on two legs), 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, causing fear and terror in the defenders of Minas Tirith.

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.

The Witch-King in The Return of the King says specifically: "Never come between a Nazgûl and its prey." Though he commands the fell beast to eat Theoden he is likely referring to himself when he says Nazgûl, not the fell beast.

Behind the Scenes
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.