A Pegasus is a winged horse capable of flight and, in Narnia, generally a Talking Beast. The first Pegasus in Narnia was Fledge, formerly known as Strawberry when he worked pulling a hansom cab in London.
Strawberry was brought into Narnia at the time of its creation by Digory and Polly, who were trying to get rid of the Empress Jadis of Charn (later known as the White Witch), along with the cab driver (who became King Frank I).
Strawberry was renamed by Aslan when he was given wings, so that he could carry Digory and Polly to the secret garden in the mountains, where Digory was charged with plucking an apple, which would grow into a tree to protect Narnia from Jadis for almost nine hundred years.
Strawberry, or Fledge (as he was known from then on) became the father of all Flying Horses in Narnia.