If you were offered the choice, would you choose a short life with eternal fame or a long life as a nameless nobody? That was the choice offered to Achilles when he was born via a prophecy given to his mother, a water nymph called Thetis. Like the majority of Greek heroes, Achilles was a demi-god, born of a union between a god and a human. Achilles was the son of Thetis by King Peleus, an unusual beginning for a demi-god as their mothers tended to be mortal women impregnated by a god.
Thetis, after hearing this prophecy, began bathing Achilles in the river Styx, which led to the underworld, in an attempt to make him invulnerable to harm. She was mostly succesful, but her neglect to consider dipping the ankle she held him by in the river's waters, left him vulnerable to injury there. His weak spot led to the use of the phrase "Achilles heel" to describe any sort of weakness.
In any case, Achilles soon rose to incredible renown as a man of valor as no mortal men knew of his vulnerability. He became the leader of the Myrmidons, a race of men whose ancestors had been transformed from ants into men long ago. He must have been a suitor of Helen at some point, for the agreement that her father made to allow men to court her forced them to assist should she be abducted. This Helen, of course, was the one whose face launched the thousand ships, starting the Trojan War. However, Achilles, following his mother's wishes, who feared that this war would result in Achilles' death, had gone into hiding disguised as a woman. His disguise was pierced by Odysseus, the most clever among the Greeks, who had King Agamemnon lay out a bunch of necklaces and a bunch of weapons among the women of the city where Achilles was hiding after gathering them together. Achilles was soon recognized as being the only "woman" who was more interested in the weapons than the jewelry.
After King Agamemnon was forced to return one of his mistresses to her father, a priest of Apollo, who had stricken the Greeks for his sacrilegious action at the behest of her father, dissension arose among the Greek warriors. Since Achilles was one of the most vocal among the Greeks in demanding that Agamemnon return the girl to her father, Agamemnon decided to revenge himself upon Achilles by taking his favorite mistress to replace the one he had just lost.
Because of this, Achilles returned to his tents in the Greek camp and no longer participated in battle against the Trojans. The tide of battle began to turn against the Greeks as a result and they were pushed back to their boats. Despite desperate entreaties by his comrades, Achilles refused to relent and help the Greeks, although he did allow his comrade and closest friend, Patroclus, to lead his troops in his place and armor. Although the Greeks rallied behind Patroclus, whom the Trojans initially took for Achilles because he lent his armor to Patroclus, his true identity is soon revealed and he is killed by Hector, who steals his armor.
When this news reached Achilles, he is devastated and swears that he will get vengeance on Hector. His mother brought him new armor fashioned by Hephaestus, the patron god of blacksmiths, himself. Thus freshly armed, he joined the battle once again and began to search for Hector. A long chase began once Achilles finds Hector, who was ultimately caught and killed by Achilles. Not satisfied with killing his foe, Achilles tied Hector's corpse to the rear of his chariot and dragged the lifeless body behind as he drove it several times around the walls of Troy. Hector's father snuck into the Greek camp at nightfall and begged Achilles to return the body. This act of devotion touched Achilles and he granted the request, which was followed by a temporary armistice for a proper mourning period.
The end of the mourning period heralds Achilles' death. He was killed shortly after by Paris, whose thrown spear was guided to Achilles' only vulnerable spot by the assistance of Apollo. Thus ended the life of the Greek hero who chose glory over a long life.