Hercules is born a demi-god. That is, one of his parents is a god (Zeus in this case) and one is a mortal. Hercules' mother is named Alcmene. Alcmene also happens to be Zeus' granddaughter via Perseus, so take that for what you will. Hercules is also one of the very few mortals to become a full god.
The biggest thing he's known for are the 12 labors. Basically, as others have said, Zeus would sleep with anything that stayed still long enough. This massively pisses of Zeus wife Hera, as would be expected, but because Hera can't take it out directly on Zeus, she tends to make life miserable for his kids and lovers. I should also point out that Zeus impersonated Alcmene's husband when he came too her, so the poor lady's really getting a bad deal all around.
In Herc's case Hera tried to prevent his mother from being able to give birth (which almost killed Alcmene). When that didn't work she sent two snakes to kill Hercules in his cradle, and Hercules strangled them.
Then, if I'm remembering right, Hera drove Hercules mad and he murdered his wife and children. To repent for that, the Oracle told him to serve King Eurystheus (Hercules' cousin) for 12 years. If he succeeds, Hercules will become immortal. Now, Herc should have been the king, but because Hera delayed his birth Eurystheus became the king. That leaves Eurystheus in debt to Hera. And so Eurystheus has Hercules do impossible things hoping to get him killed. These are the 12 labors.
The first thing he does is slay the Nemean Lion; a man-eating monster with an impenetrable hide. You normally see Hercules depicted with a lion skin. That's why.
Second, the Hydra. A venomous monster with multiple heads. If you cut one off, two more grow. Hercules dips his arrow heads in the hydra's blood to make them poisonous.
Third capture the Ceryneian Hind. This deer was extremely fast and sacred to the Goddess Artemis. Part of the plan was to get Artemis mad at Hercules, but she forgives him.
Fourth Capture the Erymanthian Boar. Another fearsome monster.
Fifth clean the Augean Stables. These were stables full of immortal animals that hadn't been cleaned for years. Herc was promised some of the cattle if he could clean the stables in a single day, and so he re-routed a river to wash them out. The cattle owner didn't want to give up the cattle, so Herc killed him.
Sixth, kill the Stymphalian birds. Man-eating metal birds with poisonous poop. Hercules scares them out of the trees and shoots them with his hydra blood arrows.
Seventh the Cretan Bull. Hercules was supposed to capture a Bull causing problems on Crete. It may have been the Minotaur's father, not sure on that one.
Eighth, steal the Mares of Diomedes. These are man-eating horses owned by a monster.
Ninth steal the girdle of Hippolyta. Hippolyta was the Queen of the Amazons.
Tenth steal the cattle from Geryon. Geryon was another monster. Hera sent flys and made rivers flood to make Hercules' job harder.
Eleventh, steal a apple of the Hesperides. These apples gave immortality and where guarded by a dragon.
Twelfth, capture and bring back Cerberus. Cerberus is the three headed dog that guards the entrance to Hades. Cerberus scares Eurystheus so much he hides and until Hercules takes the dog back to the underworld.
Hercules eventually marries again, but his bride is kidnapped by the centaur Nessus. Hercules shoots him (again with the hydra arrows), and while he dies, he convinces Hercules's wife that if she saves Nessus' blood, she can uses it as a love potion to keep Hercules faithful.
Herc follows daddy's footsteps and starts playing around, so wife uses the centaurs' blood on a vest and insists Hercules wear it. Turns out that Nessus' blood was toxic because of the Hydra venom, and Hercules gets horribly burned by it. The oracle tells him to build a funeral pyre and set himself on fire, and after he's burned alive he becomes an immortal god (although not an Olympian god).
That's basically Hercules' story, from what I remember. One final thing, his name means something like the Fame of Hera.