Short answers, pulled from the story.
Pegasus was sired by Poseidon in his role as horse-god and born from the Gorgon Medusa. He emerged along with his brother Chrysaor when Perseus beheaded Medusa at the edge of the earth near the springs of Oceanus.
Pegasus created the Hippocrene, meaning "horse spring," on Mount Helicon, the mountain sacred to the Muses. According to Antoninus Liberalis, Poseidon sent Pegasus to strike the mountain and prevent it from swelling upward with rapture at the Muses' singing.
Bellerophon was advised by Polyeidos to sleep in the temple of Athena, where the goddess appeared to him and gave him a golden bridle. The next morning he found Pegasus drinking at the Pierian spring and used the bridle to tame him.
After Bellerophon fell while attempting to reach Mount Olympus, Pegasus continued to Olympus where Zeus stabled him among the divine steeds. He was given the task of carrying Zeus's thunderbolts, attended by Astrape and Bronte.
The United Kingdom's parachute troops adopted the Pegasus insignia in 1941, showing Bellerophon mounted on the winged horse in light blue on a maroon background. The British Army credits Major Edward Seago, an East Anglian painter, with designing it in May 1942.
Pegasus Bridge is a bridge over the Caen Canal near Ouistreham in Normandy, France. British 6th Airborne Division troops captured it on the night of 5-the 6th of June 1944 during the airborne phase of the Normandy invasion, and it was named in honor of their Pegasus insignia.