Hercules

Heracles (aka Herakles) (Ηρακλής in Ancient Greek) is regarded as the greatest of ancient Greek heroes. He was born as the demigod son of Zeus and Alcemene. Heracles married Hebe the Goddess of Youth after attaining godhood. His Roman (and much more well known) counterpart is Hercules.

History
After completing his labors, Heracles joined up with Jason and the Argonauts in his quest for The Golden Fleece. Heracles later helped the Gods fight the giants (who were the children of Gaea and Tartarus) led by Alcyoneus and Porphyrion in a plot to restore the Titan rule, freed Prometheus, defeated Antaeus, lost to Dionysus in a drinking contest, and ended up mating with a half-woman, half-snake creature to reclaim his horses.

When Heracles takes Deianara as his wife, the centaur Nessus offers to carry Deianeira across a fast flowing river while Heracles swims it. However, Nessus is true to the archetype of the mischievous centaur and tries to steal Deianara away while Heracles is still in the water. Angry, Heracles shoots him with his arrows dipped in the poisonous blood of the Lernaean Hydra. Thinking of revenge, Nessus tells Deianeira to take a couple of drops of his blood if she thinks that Heracles' love will ever fade for her. Deianeira takes the blood thinking of the many ladies that would like to steal her husband.

One day while Heracles is away at war, he won a great victory and sent a messenger for his best tunic to celebrate with some men. Deianara thinking that Heracles wants his best tunic to look good for a lady, takes Nessus' blood and paints it on the tunic. Lichas, the herald, soon delivers the tunic to Heracles. However, because it is covered in the Hydra's blood from Heracles' arrow, it poisons him, tearing his skin and exposing his bones. Before he dies, Heracles throws Lichas into the sea, thinking he was the one who poisoned him (according to several versions, Lichas turns to stone, becoming a rock standing in the sea, named for him). Heracles then uproots several trees and builds a funeral pyre which Poeas (the father of Philoctetes) lights. As his body burns, only his immortal side is left. Through Zeus' apotheosis, Heracles rises to Olympus as he dies. No one but Heracles' friend Philoctetes (Poeas in some versions) would light his funeral pyre (in an alternate version, it is Iolaus who lights the pyre). For this action, Philoctetes (or Poeas) received Heracles' bow and arrows, which were later needed by the Greeks to defeat Troy in the Trojan War. The Trojan War, however, would continue until the Trojan Horse was used to defeat Troy. Philoctetes confronted Paris and shot a poisoned arrow at him. The Hydra poison would subsequently lead to the death of Paris.

As for Heracles when he is brought to Mt. Olympus, Hera ended up dropping her vendetta against him while Heracles ends up marrying Hebe, the goddess of eternal youth, also Hera and Zeus' daughter.

Accomplishments
One of the things he is most known for are the twelve labors he had to complete for King Eurystheus. These were a series of "labors" that were given to him to purify him after he was cursed with madness by Hera, causing him to murder his wife and children. His labors include:
 * 1) Slay the Nemean lion.
 * 2) Slay the Hydra.
 * 3) Capture the Ceryneian Hind, the golden stag of Artemis.
 * 4) Capture the Erymanthian Boar
 * 5) Clean the Augean Stables in a day.
 * 6) Slay the Stymphalian Birds.
 * 7) Capture the Cretan Bull.
 * 8) Steal the Man-Eating Mares of Diomedes.
 * 9) Obtain the Girdle of Hippolyta.
 * 10) Obtain the cattle that is owned by Geryon.
 * 11) Steal the apples of the Hesperides.
 * 12) Capture and bring back Cerberus.

During his labors, he found time to become one of Jason's Argonauts during their search for the The Golden Fleece.

The Lightning Theif
Heracles is mentioned by a sea nymph during a meeting underwater with Percy, he is mentioned along two others who had escaped the underworld.

The Sea of Monsters
Talking to Hermes at Camp-half blood's beach, Percy says that Heracles was his favorite hero because he had such "rotten luck" it made Percy feel better about his own plight.

The Titan's Curse
Heracles is mentioned, and Percy dreams from his point of view. Zoë Nightshade was the one who gave him the ideas for several of his labors, including how to steal the Apples of the Hesperides. Heracles, however, never even mentioned her. Once he learns how Heracles treated his friend however, he changes his mind. Percy's sword, Anaklusmos, was originally owned by Heracles which was given by Zoë Nightshade.

The Last Olympian
When Percy was getting prepared to battle he was sent on some familiar labors in which he had to complete, just like Heracles did.

The Lost Hero
Fighting with the giant Enceladus Jason said he killed the sea monster of Troy (Keto Trois) originally killed by Heracles.

Personality
Heracles was renowned for his extraordinary courage and his willingness to go to great lengths to help those he cared for. Heracles was a passionate, but somewhat unintelligent man with an emotionally untempered and impulsive nature. Percy's vision in The Titan's Curse showed him to be both confident and proud.

Abilities
As a demigod, he had these natural abilities:
 * ADHD.
 * As a child of the Big Three, his abilities were much more potent than other demigods.
 * His legendary godlike strength.
 * As a child of Zeus, he had the ability to control air and lightning.
 * Immortality (only gained after burning in a pyre)

Trivia

 * Interestingly, Heracles is absent from the series. In myths, he was made immortal upon his death with his mortal half descending to the underworld. In the series, whenever he is mentioned he is discussed as not being around. It is possible in actuality Zeus never made him a god or he simply hasn't made an actual appearance yet.
 * While as a Demigod he could throw massive boulders and monsters, like the hydra, around with little effort, it remains unknown how powerful his strength is as a full fledged god.
 * It was often said in the myths that Heracles was, and possibly still is, Zeus's favorite child.
 * Zeus fathered Heracles to serve as Olympus's mortal champion in the first Gigantomachy. It was prophesied that without a mortal to fight beside and kill the giants the Olympians would have lost the war.
 * Heracles is the half-brother and great-grandson of Perseus. See here.
 * In The Percy Jackson Series, Heracles is only called by his Roman name: "Hercules" by the Greek demigods. The only exception to this is in The Sea of Monsters, Hermes asks Percy which constellation he liked best, and Percy's reply was "Heracles" (though it was changed to Hercules in the paperback version). In The Lost Hero, Jason Grace, a Roman demigod, calls him by his Greek name: "Heracles".
 * His name, Heracles, means "Hera's Pride" in honor of Hera, who didn't show any affection towards him.
 * Since he is the great-grandson of Perseus, another son of Zeus, he can be said to be a son/legacy of Zeus.