Achilles was the son of a mortal father and an immortal mother, the nymph Thetis.
It wasn't common for immortal females to mate with mortal men, although Aphrodite is a notable exception here. Thetis wasn't happy when she realized that was her fate. Instead of getting to marry and/or mate with the head honcho, Zeus, or his slightly lower-in-status brother, Poseidon, she was paired off with a Thessalian king named Peleus.
Peleus had proven himself honorable to the gods and deserving of a reward, but not of an unambiguous sort. Poseidon and Zeus had turned against Thetis because they had learned that whoever fathered her son, would prove to be much less of a man than the son.
The gods couldn't handle it. They knew from personal experience of a couple of generations of similarly ever-enhanced genetics just how dangerous a son could be. Zeus had overthrown his father, and his father had overthrown his own by an act of castration. Neither the promiscuous Zeus nor Poseidon had any desire to be unmanned and Zeus was quite happy to be king of the heavens and head god.
So Peleus was selected as husband for Thetis partly because he deserved an honor, but also because the gods really didn't care what happened to him.
Thetis wasn't happy. She was a shape-shifter and tried to get away from Peleus, but Peleus kept hanging on. Perhaps Thetis was impressed with his endurance. Perhaps she got tired. Whatever the reason, Thetis agreed to marry Peleus and produced a son and possible heir, Achilles.
In a moment of maternal bonding, to keep her new-born infant safe, Thetis decided to make him immortal. There were various ways of doing this, but to observers they seemed likely to kill the child. Thetis chose to dunk her baby boy in the Underworld's River Styx (also note that in other non-Achilles' heel versions of Thetis' actions, she thrust him into boiling water, or, like Demeter, tried to make the baby immortal by putting him in fire). So that he wouldn't drown or float away, she held him by his left ankle. Why she didn't take turns and hold him for a second dunk by the right ankle we'll never know, but that's what Thetis did and the result was that Achilles was impervious to assault, except in the one spot that hadn't been made immortal -- his heel or more specifically, his Achilles' heel.
Although this is a sort of back story, it doesn't fit too well with the details about Achilles' career. In the stories, Achilles suffers wounds elsewhere on his body, despite the early baptism. Writers tend not to specify that an arrow pierced his ankle. Even when his ankle is named, it takes a bit of a leap to believe that as great a man as Achilles would be killed by such a relatively insignificant wound.
Also see: "Achilles' Heel: The Death of Achilles in Ancient Myth," by Jonathan Burgess Classical Antiquity , Vol. 14, No. 2 (Oct., 1995), pp. 217-244.
Image: CC Flickr User ketrin1407
