Here's one version of Perseus' story, as told by Thomas Bulfinch in The Age of Fable. For more variations on the story, visit Theoi.com.
Perseus was the son of Jupiter and Danae. His grandfather Acrisius, alarmed by an oracle which had told him that his daughter's child would be the instrument of his death, caused the mother and child to be shut up in a chest and set adrift on the sea. The chest floated towards Seriphus, where it was found by a fisherman who conveyed the mother and infant to Polydectes, the king of the country, by whom they were treated with kindness. When Perseus was grown up Polydectes sent him to attempt the conquest of Medusa, a terrible monster who had laid waste the country. She was once a beautiful maiden whose hair was her chief glory but as she dared to vie in beauty with Minerva, the goddess deprived her of her charms and changed her beautiful ringlets into hissing serpents. She became a cruel monster of so frightful an aspect that no living thing could behold her without being turned into stone. All around the cavern where she dwelt might be seen the stony figures of men and animals which had chanced to catch a glimpse of her and had been petrified with the sight. Perseus, favoured by Minerva and Mercury, the former of whom lent him her shield and the latter his winged shoes, approached Medusa while she slept and taking care not to look directly at her, but guided by her image reflected in the bright shield which he bore, he cut off her head and gave it to Minerva, who fixed it in the middle of her AEgis.
After the slaughter of Medusa, Perseus, bearing with him the head of the Gorgon, flew far and wide, over land and sea. As night came on, he reached the western limit of the earth, where the sun goes down. Here he would gladly have rested till morning. It was the realm of King Atlas, whose bulk surpassed that of all other men. He was rich iii flocks and herds and had no neighbour or rival to dispute his state. But his chief pride was in his gardens whose fruit was of gold, hanging from golden branches, half hid with golden leaves. Perseus said to him, "I come as a guest. If you honour illustrious descent, I claim Jupiter for my father; if mighty deeds, I plead the conquest of the Gorgon. I seek rest and food." But Atlas remembered that an ancient prophecy had warned him that a son of Jove should one day rob him of His golden apples. So he answered, "Begone! or neither your false claims of glory nor parentage shall protect you;" and he attempted to thrust him out. Perseus, finding the giant too strong for him, said, "Since you value my friendship so little, deign to accept a present;" and turning his face away, he held up the Gorgon's head. Atlas, with all his bulk, was changed into stone. His beard and hair became forests, his arms and shoulders cliffs, his head a summit, and his bones rocks. Each part increased in bulk till be became a mountain, and (such was the pleasure of the gods) heaven with all its stars rests upon his shoulders.
[See our guide to Perseus and Andromeda for the story of the hero defeating the sea monster and winning Andromeda as his wife.]
Then Perseus returned to Seriphus to King Polydectes and to his mother Danae and the fisherman Dicte. He marched up the tyrant's hall, where Polydectes and his guests were feasting. "Have you the head of Medusa?" exclaimed Polydectes. "Here it is," answered Perseus, and showed it to the king and to his guests.