Pegasus is the best known winged horse (pterippus) of Greek and Roman mythology.
Pegasus was 'born' when Perseus decapitated Medusa, rising from her blood as it seeped into the ground (in some versions, it's a mixture of blood and seafoam). He is said to be the son of Poseidon, who approached Medusa while he was in the form of a horse{1}; it was a consequence of this mating that caused Athena to transform Medusa into a monster. Pegasus has a brother, Chrysoar, who arose in the same way.
After he had helped Perseus with a few adventures (killing Cetus, rescuing Princess Andromeda), Athena (AKA Minerva) caught him with a golden bridle (named Chalintis) and took him to Mount Helicon, where she left him in the care of the Muses. On one occasion, the muses started to sing; they sang so beautifully that the mountain started to rise towards the heavens in ecstasy. Pegasus struck the ground with his hooves, stopping the mountain from rising, and causing the springs of Aganippe and Hippocrene to burst forth{2}. The rising of these springs marked the birth of both wine and art.
But Pegasus' happy days with the Muses were not to last... {3} Athena gave him to the hero Bellerophon, who needed him to kill the dreaded Chimera, which was terrorizing Lycia. They beat the Chimera, and went on to battle many other monsters, including the infamous Amazons. Bellerophon returned in triumph, and married King Iobates' daughter, Philonoe. They had some children, Bellerophon became king, and they were generally happy -- until Bellerophon decided to try to fly to Mount Olympus, in an attempt to become a god. Zeus didn't like this, and sent a fly to bite Pegasus. Pegasus bucked, sending Bellerophon plummeting to the ground, where he wandered, crippled and confused, for the rest of his days{4}.
While Bellerophon was not welcome in Olympia, Pegasus was. He was entrusted with bringing lightening and thunderbolts to Zeus; in fact, Pegasus' hooves may have been the cause of the thunder. He sometimes helped Eos (Aurora) with her morning drive across the sky, creating the dawn, and Apollo (Phoebus) with his evening journey, causing nightfall. Zeus honoured Pegasus by giving him his own constellation.
In his later life, Pegasus took a wife, Euippe (or, some say, her sister Ocyrrhoe), who was the daughter of the good centaur Cheiron, by whom he had two children, Celeris and Melanippe. This family is the origin of the pterippi.
If you should happen to want to use Pegasus in the plural (you shouldn't. By now you should have caught on to the distinction between Pegasus and pterippus), you can use pegasi. Every time I hear it, I think of a fantasy story where the Greek gods have invented cloning... "Look Zeus! Pegasi! Pegasi for all!"
1 In some versions, he was actually in the form of a centaur. Poseidon, besides being the ruler of the sea, is the god and guardian of horses.
2 This association with the muses, and the springs, has made Pegasus a symbol of the arts, especially poetry. Drinking from the spring Hippocrene is supposed to make you an excellent poet.
3 The famous demigod and hero Bellerophon had gotten himself in trouble; he had been working for King Proteus, when the Queen, Antea tried to seduce him. He refused, and she turned on him, demanding that King Proteus put him to death. Proteus fobbed him off on King Iobates; Iobates was surprised that Proteus would want him to kill a follower that he had praised so highly, and so instead of putting Bellerophon to death, he sent him off to kill the Chimera. A soothsayer (Poluidus) informed Bellerophon that the only way he could kill the Chimera would be if he rode Pegasus. Bellerophon could not catch Pegasus himself, but after he spent the night (for once, no sex was involved) in the temple of Athena, she gave him Chalintis (the magic bridle, remember?).
4 Some versions say that Pegasus would not obey Bellerophon's order "to Olympus!", and bucked him off without any biting insects to prompt him. Some versions say that Bellerophon fell to his death.
http://members.tdn.com/nwbhc/breeds/equines/longears/pegasus.htm
and
http://www.thanasis.com/pegasus.htm
are good sites for more Pegasus related stuff.
http://www.seds.org/Maps/Stars_en/Fig/pegasus.html
is a good site about the constellation.