The term originated from this Greek story: there was this guy named Oedipus who, at birth, was destined by fate to kill his father (who also happened to be the King of Thebes) and marry his mother. The people in Thebes thought that this was a pretty big deal, so when Oedipus was an infant, he was sent away. As an adult, Oedipus returns to Thebes only to - you guessed it - kill his dad and marry his mom.
By the way, the reverse condition (women who seek to kill their mother and marry their father) is called the Elektra complex.
* Achilles' heel: Achilles was born to a goddess mother, Thetis, and a human father, Peleus. Thetis did not like the fact that her son was part mortal, so she was told to dip him into the River Styx to cleanse him of his mortality and render him invincible. She did this, but held him by his heel as she dipped. His heel was left untouched by the water, and so that remained the only part of him that was not invincible. Years later when he was in battle in the Trojan War, Achilles' enemies took advantage of his weak spot and shot a poisonous arrow directly into his heel, and he died.
We now use this term to refer to someone's weakest spot, the cause of his or her downfall. For instance, if ice cream is your Achilles' heel, then the one way someone can get you to do something for him or her is to offer you ice cream.
* A Herculean task: Hercules is actually the Roman name of the Greek hero, Heracles. Heracles is famous for his unbelievable strength. Throughout his life, Heracles was also challenged to many physical tasks (such as defeating the many-headed Hydra). Each and every time, Heracles successfully accomplished his "labors," all because of his god-like strength....