In the myths, Hercules was the son of Jupiter (Zeus Pater = Jupiter), the supreme god of Olympus, and of Alcmene, a mortal married woman. Juno (Hera, to the Greeks) the wife of Jupiter, hated Hercules because he was the most famous and successful of Jupiter's numerous illegitimate progeny. Being a goddess, she could, of course, see the future, so she delayed his birth and then, when he was finally born, sent two large snakes to kill him as he lay in his crib. But, even as a child, Hercules had such strength that he strangled the snakes, one in each hand. That image of the infant Hercules strangling the snakes was extremely popular in early ancient Rome, because Rome saw itself as a precocious infant state conquering dangerous local enemies. The fact that in some way this might be done in spite of the will of the gods added an interesting frisson, which Rome's jaded upper classes might appreciate.
Despite Juno's early efforts, Hercules lived to become the famous hero that he still is today. She eventually did cause him a lot of grief: she drove him mad, and in one of his manic fits, he murdered his wife and children, thinking he was slaying his cousin and rival, Eurystheus, and his family -- which would, of course, have been exemplary heroic behavior in the moral structure of his day. (Because of this incident and others, Hercules' would nowadays probably be diagnosed as having "bipolar disorder", formerly called manic-depression. The close relationship of "heroism", in the ancient sense, to madness was clearly one of the lessons the ancient mythographers were trying to convey.)
To expiate his crime of familiacide, Hercules submitted himself to his Eurystheus, his enemy and the King of Mycenae. Eurystheus required him to perform a series of nearly impossible tasks known in classical mythology as the "Twelve Labors of Hercules". To re-acquire mental balance, Hercules had to:
1. Kill the Nemean Lion
2. Destroy the Lernean Hydra
3. Capture the Cerynean Hind (a stag with golden horns)
4. Capture the wild boar of Erymanthus
5. Clean the Augean stables in a single day
6. Drive away the Stymphalian birds
7. Capture the Cretan bull
8. Capture the man-eating horses of Diomedes, King of Thrace
9. Bring back the girdle of Hippolyta, Queen of the Amazons
10. Capture the cattle of Geryon, a monster who had three
bodies joined at the waist
11. Bring back the Golden Apples of the Hesperides
12. Capture Cerberus, the three-headed dog of Hades
It was in the aftermath of labor number ten that Hercules showed up in Rome. Exhausted from slaying Geryon, Hercules drives the captured cattle across the Tiber River and settles down for a nap. While he's asleep, the local bad guy, Cacus, a fire-breathing son of Vulcan, steals four bulls and four cows from Hercules' newly liberated herd, and drags them by their tails into his cave. This backwardness is purposeful: knowing that Herc has more brawn than brains, Cacus fools Hercules by leaving a backward trail. Initially, the ruse works, but one of the bulls outside bellows for his missing truelove and she moos a response from inside the cave. Hercules slowly figures out the truth and then goes in after Cacus and the rustled cattle. After a struggle, in which Hercules is "half singed, half stifled" by Cacus' flames and fumes, Hercules throttles him, ties his arms and legs in knots and plucks out his eyes. And that was the end of Cacus.
All of this happened, of course, before Aeneas and his kin arrive in Latium in about 1200 BC. Eurystheus, who was Hercules' taskmaster, was a semi-mythological Mycenaen king, and the Mycenaen bronze age culture was already pretty well spent by the time of Aeneas' voyage. When Aeneas finally reaches Italy, he finds a pre-existing coastal temple celebrating the local exploits of Hercules. Evander, a Greek colonial king who lived in the area and who had entertained Hercules, had raised another altar and a statue of Hercules. According to some versions of the myths, Evander was still alive when Aeneas arrived and was eventually an ally of Aeneas.
The early presence of Hercules' in Rome was thus a staple of local mythology from Rome's earliest days, and Evander's early temple in his honor was in the Forum Boarium -- an obvious association of the cattle legend with Rome's cattle market, which is what "Forum Boarium" means. Although literary descriptions of the location of the earliest small altar in the market are extremely precise, no remains have ever been found, and, similarly, no remains of Evander's statue of Hercules have been found: both were reportedly destroyed in Nero's fire and the "urban renewal" that followed. Also in the Forum Boarium, there are remains under the church of Santa Maria in Cosmedin (where the "Mouth of Truth" stands) of a later temple dubiously identified as dedicated to Hercules. Across the street is the round temple, long mis-identified as belonging to Vesta, which, although often modified and rebuilt and perhaps even moved, is now thought to be a temple of Hercules Victor. The new glass windows and doors installed in 2000 to protect interior paintings finally make a reality of the myth that dogs and flies wouldn't enter the temple because Hercules left his famous club leaning against the doorframe.
Strange parallels: One of the most common titles of Hercules was "Son of God", by virtue of his having Jupiter as father and an earthly mother. He was viewed as a savior of the oppressed, but his people, the Heracleidae, were persecuted after his death. Legend held that he had raised two people from the dead. He suffered an excruciating death, descended into Hades for three days, and returned after his death to show his closest friends that he was still alive and using his wounds to prove his identity, before ascending bodily into heaven. Hercules was deeply admired among the people, a fact not ignored by early Christian leaders and writers. Some historians argue that the image of Hercules still exists (although much softened) in the familiar face of Jesus portrayed as a white European with straight hair. Helenistic Christians replaced the original image of Jesus with the darker complexion and tightly curled hair common among 1st century Palestinian Jews. But the similarities were not restricted to Hercules and Jesus: virgin births, re-animations of the dead, sacrificial deaths, mystical journeys, resurrections, etc., were also among the grab-bag characteristics of Romulus, Mithras, Zoroaster, Isis, and other Mediterranean "founders"
Delusions: The iconography of Hercules features the lion skin (the hide of his first-labor adversary) and the oaken club, which was Hercules' weapon of choice. If you see a statue in Rome that has both of these "attributes" you are looking at Hercules -- or maybe at Commodus or at Mussolini or at any one of a number of historical delusional Hercules pretenders. They just keep popping up, even after all these centuries.
The Labors of Hercules -- About.com: http://ancienthistory.about.com/homework/ancienthistory/library/weekly/aa032701a.htm
Herakles in Rome: http://www.dcs.ed.ac.uk/home/me/rome.html
Hercules and Cacus, from Vergil's Aeneid: http://classics.mit.edu/Virgil/aeneid.8.viii.html
The Herakles Project from UPenn's ForvmAntiqvvm: http://www.sas.upenn.edu/~ekondrat/heraklesmain.html
Heracles/Hercule from Le Grenier de Clio: http://www.chez.com/clio/mytho/heracles.htm