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Prometheus, along with his brother Epimetheus, were two of the four sons of the Titan Iapetus (a son of Uranus--Heavens-- and Gæa--Earth--, and thus elder brother of Cronus) and the Oceanide Clymene (or Asia, a daughter of Ocean and Tethys in both cases). Thus they were cousins of Zeus, who was the son of their uncle Cronus. Their brothers were Menoetius (who was so proud and rude that Zeus struck him with lightning and plunged him into Tartarus, as he did with his father and all the Titans) and Atlas (see Hesiod's Theogony, 507, sq). Prometheus was as shrewd as his brother Epimetheus was clumsy.
Prometheus became the father of Deucalion, the first man, with his wife Celæno, or Clymene (in traditions where she is not mentioned as his mother). He was also sometime given as the father of Hellen, the ancestor of all Greek tribes, in the place of his son Deucalion. In fact, in some traditions, he was even said to have manufactured the first men from clay. He was the benefactor of mankind that he protected from the jealousy of Zeus and the blunders of his brother.
During a solemn sacrifice, Prometheus deceived Zeus by asking him to pick his share between two shares of the ox he had slaughtered, the other being for men. In one, Prometheus had put the meat and entrails hidden behind the belly of the animal, and in the other, the bare bones hidden behind white fat. Zeus chose the later, leaving the former to men. But when he saw what he had taken and what he had lost, he became jealous of men and withdrew them the fire. But Prometheus came to the rescue of mankind in stealing fire from Hephæstus' forge.
In reprisal, Zeus asked Hephæstus and Athena to manufacture, with a bit of help from all other gods, a creature that would be the source of all evils for men, a woman (the first one indeed), named Pandora (a name meaning in Greek "the all-encompassing gift"). Pandora was so beautiful that, no sooner had Epimetheus seen her that, forgetting the order from his brother Prometheus never to accept a gift from Zeus, he fell in love with her and made her his wife. As soon as she was on earth, Pandora, seeing a jar that was there and devoured by curiosity, opened it. Halas ! The jar was holding all the evils, and they immediately spread over mankind. Another tradition presents the jar as Pandora's wedding gift to Epimetheus, that was holding all goods. But, because Pandora opended it, all goods escaped and returned to the gods, except hope, that was at the bottom and could not escape before Pandora put the lid back on the jar (see Hesiod's Theogony, 507-616 ; Works and Days, 42-105).
Prometheus' punishment was to be chained to a huge rock in Caucasus where an eagle would come every day to eat his liver, that would reconstitue immediately. And Zeus vowed never to unchain Prometheus from his rock. Yet, when Heracles happened to pass by, he killed the eagle and unchained Prometheus, who told him how to get the Golden Apples from the garden of the Hesperides. Zeus was proud of this deed of his son, but, so as not to renege on his vow, he ordered Prometheus to always wear a ring that would be made out of the steel from his chains and a piece of the rock he had been tied to.
Toward that time, the Centaur Chiron, wounded by Heracles' arrows, wished to die. But, because he was immortal, he could do so only if he could find some mortal that would take over his immortality. Prometheus accepted the deal, and thus became immortal, with Zeus' blessing at last, because, owing to his gift of prophecy, he had warned him that, if he had a son with the Nereid Thetis, with whom he was then in love, this son would become stronger than him and unseat him (no longer courted by the gods, Thetis later became Achilles' mother).
It is also Prometheus who warned his son Deucalion, who had by then married Pyrrha, the daughter of Epimetheus and Pandora, of the impending flood Zeus was planning to destroy mankind, and gave him the means to escape the disaster.

Plato puts in the mouth of Protagoras a reworked myth of Prometheus and Epimetheus in the dialogue that bears his name (Protagoras, 320d-322d), in which Pandora is out and Epimetheus' clumsiness is responsible for man's weakness, while Prometheus is responsible for bringing them fire and the arts ("technai") stollen from Hephæstus and Athena, before Zeus intervenes to instruct Hermes to bring them justice and a sense of shame that would make it possible for them to associate and live in cities. Prometheus is also mentioned in the final myth of last judgment in the Gorgias as responsible for the ability that men once had to know the hour of their death ahead of time (Gorgias, 523d-e). Again, in the myth of the Statesman about the golden age of Cronus and the time when the earth is left to itself, in a reference to god given gifts to men, Prometheus is mentioned as the one who gave men fire (Statesman, 274c). And in the Philebus, Socrates attributes to "some sort of Prometheus" the godly gift of "a most dazzling fire" that allows men to partake in the knowledge of the one and the many (Philebus, 16c).