In Greek Mythology, dead mortals generally ended up in one of three places in the underworld, known collectively as Hades. The virtuous went to Elysium. Those who were neither virtuous nor evil went to the Plain of Asphodel. The truly wicked were sent to Tartarus. Its punishments were normally reserved for fallen deities. There are several notable exceptions to this, however, whose names and punishments were so iconic that they have made their way into the English language.
The fates and punishments of mortals were judged by three men whose good judgment in life earned them their places in the afterlife. The division of labour differs with the sources but generally speaking Rhadamanthus judged Asians, Aeacus judged those from Europe, and Minos was either the tiebreaker or the man to whom the other two forwarded difficult cases. Some mortals appealed directly to Hades or Persephone (the god and goddess of the Underworld, respectively) for special favours, but direct intervention by the gods was rare.
Tantalus was guilty of two major sins. His first was to share the secrets he learnt at Zeus' table with others. His second was to serve the gods a stew made out of his own children in an attempt to trick them into eating the flesh of mortals, something they are not allowed to do. As punishment, Tantalus is tied to a fruit tree, submerged in a river. Though the water and the fruit are close, they are eternally just out of his reach. This is why Tantalus is the root of the word "tantalise."
Sisyphus was a sneaky fellow and a crook. He outsmarted the gods and escaped death twice before he finally died of old age. For the crime of telling Zeus' enemy where he was hiding said enemy's daughter, Sisyphus must roll a huge rock up a steep hill. When he is able to push it over the other side, his punishment will end. Unfortunately, the rock always slides back down the hill just as it nears the top. Today, a "Sisyphean task" is one that is difficult to the point of futility.
Other Notable Inmates
For the crime of killing their husbands on their wedding night, the Danaids, a group of 50 sisters married to their 50 male cousins, must forever carry water from one place to another in jars full of holes. Ixian, who tricked his father-in-law-to-be into falling into a pit of burning embers, spends eternity tied to a flaming wheel that is rolled around Tartarus. Tityus, who was slain by Apollo and Artemis, must spend eternity having his liver pecked out by vultures for attempting to rape their mother, Leto. His punishment is similar to that given to Prometheus, whose liver is also eaten, only to continuously grow back.