The original flying horse named Pegasus is found in Greek mythology. The story goes that he was created from the blood spilled when Medusa, that evil Gorgon who turned people to stone by looking at them, was killed by Perseus. He cut off her head and the blood that spurted from her neck as it mingled with the sea foam gave birth to Pegasus (which explains how his father was Poseidon, god of the ocean).
Pegasus is particularly famous for helping the Greek hero Bellerophon to destroy the Chimera, a dangerous creature, a composite of a lion, a goat, and a serpent. It seems that the hero became rather arrogant after this feat, not surprising! Able to ride Pegasus both on land and in the air made him feel invincible and he tried to fly up to Mount Olympus, the home of the gods. Pegasus apparently would have none of this and threw Bellerophon from his back, for which he gained the favor of the gods. Zeus rewarded Pegasus with a stable in Mount Olympus and was given the task of collecting thunderbolts for the god - not a bad job!
Pegasus was not immortal, though, and when he died he was turned into a constellation of stars in the sky, achieving a different kind of immortality. Also, Pegasus has lived on as the name of flying horses in mythologies since then, even to this day.