Eris, the goddess of strife, was not invited to the wedding of Peleus and Thetis. She came anyway, and she threw a golden apple into the middle of the wedding. Inscribed on the apple was a message. It read "To the fairest." Immediately, the apple was claimed by Hera, Athena and Aphrodite. They all asked Zeus to decide on who should receive the apple. Zeus knew how much trouble he would be in if he decided on one, because the other two would have grave revenge. So Zeus descended to Mount Ida where Paris was farming and asked him to be the Judge.
Paris, being a mortal, could not decide. However, each of the three goddesses decided to make it easier for him. They would each offer him gifts, and he would get the gifts form the goddess he chose.
Hera offered Paris power. She offered to give him all of Asia, and great power. He thought this offer was great, but he decided to hear the other offers first before deciding.
Athena offered him great wisdom, and great luck in battle. He would be the best strategist in the world. He loved this idea, but he waited to hear Aphrodite's offer.
Aphrodite offered him two things. The first was his body, and the second was the love of the most beautiful woman in the world, Helen. Since Paris's first love was women, he decided to pick Aphrodite's offer. Hera and Athena vowed vengeance.
Paris soon went home to Troy after that, and with Aphrodite's help, he managed to send a fleet of ships, break into Menelaus's palace in Greece and kidnap Helen. He also took a lot of treasure with him.
As Helen was the most beautiful woman in the world, all of her suitors were the most powerful people in Greece. In order for peace to be kept when Helen chose a suitor, all other suitors must vow to keep Helen as the wife of whom she chose. So when Helen chose Menelaus, all of the other suitors had to agree that if anyone tried to kidnap her, they would try to get her back. So, when Paris kidnapped her, all of Greece declared war on the city of Troy.
These actions of Paris and Aphrodite started the Trojan war. Paris fights, but he is mostly out of legend until Hector is killed by Achilles. While Achilles and his allies bring Hector's body back into Troy for a funeral, Paris take a bow and arrows and shoots it at Achilles. Apollo guides his arrow so that it hits Achilles's foot, in the famous Achilles tendon. Achilles dies. Paris is soon killed afterwards in the war.