Here's one version of the story of Heracles' Twelve Labors, as told by Thomas Bulfinch in The Age of Fable. For more variations on the story, see Apollodorus' version of the tale and Diodorus Siculus' version.
HERCULES was the son of Jupiter and Alcmena. As Juno was always hostile to the offspring of her husband by mortal mothers, she declared war against Hercules from his birth. She sent two serpents to destroy him as he lay in his cradle, but the precocious infant strangled them with his own hands. He was, however, by the arts of Juno rendered subject to Eurystheus and compelled to perform all his commands. Eurystheus enjoined upon him a succession of desperate adventures, which are called the "Twelve Labours of Hercules." The first was the fight with the Nemean lion. The valley of Nemea was infested by a terrible lion. Eurystheus ordered Hercules to bring him the skin of this monster. After using in vain his club and arrows against the lion, Hercules strangled the animal with his hands. He returned carrying the dead lion on his shoulders; but Eurystheus was so frightened at the sight of it and at this proof of the prodigious strength of the hero, that he ordered him to deliver the account of his exploits in future outside the town.
His next labour was the slaughter of the Hydra. This monster ravaged the country of Argos, and dwelt in a swamp near the well of Amymone. This well had been discovered by Amymone when the country was suffering from drought, and the story was that Neptune, who loved her, had permitted her to touch the rock with his trident, and a spring of three outlets burst forth. Here the Hydra took up his position, and Hercules was sent to destroy him. The Hydra had nine heads, of which the middle one was immortal. Hercules struck off its heads with his club, but in the place of the head knocked off, two new ones grew forth each time. At length with the assistance of his faithful servant Iolaus, he burned away the heads of the Hydra, and buried the ninth or immortal one under a huge rock.
Another labour was the cleaning of the Augean stables. Augeas, king of Elis, had a herd of three thousand oxen, whose stalls had not been cleansed for thirty years. Hercules brought the rivers Alpheus and Peneus through them, and cleansed them thoroughly in one day.
His next labour was of a more delicate kind. Admeta, the daughter of Eurystheus, longed to obtain the girdle of the queen of the Amazons, and Eurystheus ordered Hercules to go and get it. The Amazons were a nation of women. They were very warlike and held several flourishing cities. It was their custom to bring up only the female children; the boys were either sent away to the neighbouring nations or put to death. Hercules was accompanied by a number of volunteers, and after various adventures at last reached the country of the Amazons. Hippolyta, the queen, received him kindly, and consented to yield him her girdle, but Juno, taking the form of an Amazon, went and persuaded the rest that the strangers were carrying off their queen. They instantly armed and came in great numbers down to the ship. Hercules, thinking that Hippolyta, had acted treacherously, slew her, and taking her girdle made sail homewards.
Another task enjoined him was to bring to Eurystheus the oxen of Geryon, a monster with three bodies, who dwelt in the island of Erytheia (the red), so called because it lay at the west, under the rays of the setting sun. This description is thought to apply to Spain, of which Geryon was king. After traversing various countries, Hercules reached at length the frontiers of Libya and Europe, where he raised the two mountains of Calpe and Abyla, as monuments of his progress, or, according to another account, rent one mountain into two and left half on each side, forming the straits of Gibraltar, the two mountains being called the Pillars of Hercules. The oxen were guarded by the giant Eurytion and his two-headed dog, but Hercules killed the giant and his dog and brought away the oxen in safety to Eurystheus.
The most difficult labour of all was getting the golden apples of the Hesperides, for Hercules did not know where to find them. These were the apples which Juno had received at her wedding from the goddess of the Earth, and which she had entrusted to the keeping of the daughters of Hesperus, assisted by a watchful dragon. After various adventures Hercules arrived at Mount Atlas in Africa. Atlas was one of the Titans who had warred against the gods, and after they were subdued, Atlas was condemned to bear on his shoulders the weight of the heavens. He was the father of the Hesperides, and Hercules thought might, if any one could, find the apples and bring them to him. But how to send Atlas away from his post, or bear up the heavens while he was gone? Hercules took the burden on his own shoulders, and sent Atlas to seek the apples. He returned with them, and though somewhat reluctantly, took his burden upon his shoulders again, and let Hercules return with the apples to Eurystheus.