She was originally known as the fairest woman in Greece, but then when Zeus came down and cheated on Hera, his wife, with Medusa in Hera's temple, Hera became outraged. Since she was unable to touch Zeus, since he was king, she instead turned Medusa and her two other sisters, known as the Gorgons, into the most foul looking creatures in the world. While Medusa was mortal, her sisters were not, but Medusa had it worse off. Whenever she would meet somebody's gaze, she would turn them to stone from her ugliness. From this, she was put into hiding in a cave with her sisters. She was there for many years, when Perseus, out on a quest (another story), went and cut her head off in her and her sisters' sleep. He was not turned to stone because he had a shield granted to him by Athena, goddess of wisdom and warfare, which would reflect the image of Medusa so that he would not have to look her in the eyes. When he cut off her head, her blood created Pegasus, the horse with wings. With his flying sandals, also granted by Athena, he was able to escape the Gorgons without being noticed, and could fly over the sea. When he reached land and put the head down, the snakes from her hair detatched themselves, and created the first snakes on land. Perseus put the head into a bag, and returned to the king which had sent him on his quest. When the tyrannical king asked to see the head as proof of his completion of the quest that should not have been able to be completed, the king met the gaze of the glazed eyes of Medusa, and was turned to stone, therefore proving that her magical curse was still in affect. The rest is the story of Perseus. If you want to know more about that, then ask a different question.
