King Polydectis wanted to have an relationship with Danae (Perseus mum) . But he didn't want Perseus to get in the way so he told him that if he went a killed medusa and came back with her head, he said he wouldn't be with her mothers. Polydectus thinking that he wouldn't come back alive.
Athena Goddess of wisdom and Hermes Messenger of the gods told him he had to go to the Graiae (sisters of the Gorgons) he took one tooth and one eye from them not giving it back until they provided him with winged sandals, cap of Hades which made him invisible, sword/sickle, and a bag to put medusa's head in, also got a shield from Athena.
In Medusa's lair on the ends of the Earth Perseus crept into her cave being invisible from the cap of Hades and used Athena's shield to slice medusa off while she was sleeping. Whoever even looked at her turned to stone because of how hideous she was.
From the blood that ran from Medusa's neck out sprung from her neck Pegasus and Chrysaor her two offspring of Poseidon
As Perseus returned home back on the island of Seriphus, he found out that Polydectus was abusing Danae while Perseus was absent. So Perseus showed the head of Medusa to him and he turned to stone.
Perseus returned Medusa's head to Athena and she placed it on her shield. He also returned the sandals, cap and sword to Hermes.