There are two stories of the birth of the centaur race: in the first, Ixion (Ἰξίων), king of the Lapiths (Λαπίθαι), and Nephele (Νεφέλη), a cloud nymph Zeus created in the image of Hera to trick Ixion to test his integrity after displaying his lust for Hera during a feast as a guest of Zeus, fathered the Centaurs with the nymph after failing said test. The Lapiths were a mythical, pre-Hellenic tribe whose home was in Thessaly, in the valley of the Peneus and on the mountain Pelion.
Because of the antiquity of the tribe, Ixion's parents are a little hard to determine; sources attest to Ares, Apollon, or Leonteus, or Antion and Perimele. In another version of the myth, Ixion and Nephele had only one son, Centaurus, who was deformed and hunched over. He took to the wilds his father loved, and eventually mated with the mares of a race of horses that lived in Magnesian. From this union came the Centaurs. In all versions of the myth, Ixion was expelled from Olympus and blasted with a thunderbolt for failing his test. Zeus then ordered Hermes to bind Ixion to a winged fiery wheel that was always spinning. Therefore, Ixion is bound to a burning solar wheel for all eternity, at first spinning across the heavens, then into Tartaros. From Pindar's Second Pythian Ode:
"To your benefactor return ever with kind dealing rendered. He learned that lesson well. By favor of the sons of Kronos, he was given a life of delight but could not abide blessedness long; in his delirious heart he loved Hera, dedicated to the high couch of Zeus. That outrage hurled him into conspicuous ruin. He was a man and endured beyond all others distress full merited. Two sins flowered to pain in his life: a hero, he first infected the mortal breed with kindred bloodshed, not with-out treachery; also, in the great secret chambers of Zeus he strove to ravish the Queen. A man should look at himself and learn well his own stature.
The coupling unnatural brought accumulation of evil on him, even in success; it was a cloud he lay with, and he in his delusion was given the false loveliness. A phantom went in the guise of that highest daughter of Uranian Kronos; a deceit visited upon him by the hands of Zeus, a fair evil thing. Zeus likewise wrought the crucifixion on the wheel, Ixion's bane; and, spinning there, limbs fast to the ineluctable circle, he makes the message a thing that all may know.
But she, graceless, spawned a child of violence. There was none like her, nor her son; no honor was his portion in the usage of god or man. Nursing him, she named him Kentauros, and he coupled with the Magnesian mares on the spurs of Pelion; and a weird breed was engendered in the favor of either parent: the mare's likeness in the parts below, and the manlike father above. "
"After this, when Peirithous was about to marry Deidameia, he asked Theseus to come to the wedding, and see the country, and become acquainted with the Iapithae. Now he had invited the Centaurs also to the wedding feast. And when these were flown with insolence and wine, and laid hands upon the women, the Lapithae took vengeance upon them. Some of them they slew upon the spot, the rest they afterwards overcame in war and expelled from the country, Theseus fighting with them at the banquet and in the war. Herodorus, however, says that this was not how it happened, but that the war was already in progress when Theseus came to the aid of the Lapithae and that on his way thither he had his first sight of Heracles, having made it his business to seek him out at Trachis, where the hero was already resting from his wandering and labours; and he says the interview passed with mutual expressions of honor, friendliness, and generous praise."
"You used to think that the race of centaurs sprang from trees and rocks or, by Zeus, just from mares – the mares which, men say, the son of Ixion covered, the man by whom the centaurs though single creatures came to have their double nature. But after all they had, as we see, mothers of the same stock and wives next and colts as their offspring and a most delightful home; for I think you would not grow weary of Pelion and the life there and its wind-nurtured growth of ash which furnishes spear-shafts that are straight and at the same time do not break at the spearhead. And its caves are most beautiful and the springs and the female centaurs beside them, like Naïads if we overlook the horse part of them, or like Amazons if we consider them along with their horse bodies; for the delicacy of their female form gains in strength when the horse is seen in union with it. Of the baby centaurs here some lie wrapped in swaddling clothes, some have discarded their swaddling clothes, some seem to be crying, some are happy and smile as they suck flowing breasts, some gambol beneath their mothers while others embrace them when they kneel down, and one is throwing a stone at his mother, for already he grows wanton. The bodies of the infants have not yet taken on their definite shape, seeing that abundant milk is still their nourishment, but some that already are leaping about show a little shagginess, and have sprouted mane and hoofs, though these are still tender.
How beautiful the female centaurs are, even where they are horses; for some grow out of white mares, others are attached to chestnut mares, and the coat’s of others are dappled, but they glisten like those of horses that are well cared for. There is also a white female centaur that grows out of a black mare, and the very opposition of the colours helps to produce the united beauty of the whole."
It is said that the constellation Centaurus represents Centaurus himself; not because he did something heroic, but because Centaurus was the first person to group stars into constellations and taught others how to read them. The constellation might simply have been Centaurus putting a picture of his children (or, in some versions of the myth--where he is a Centaur--himself) in the sky, possibly to guide the Argonauts on their mission to secure the golden fleece.
There were many famous centaurs in Hellenic mythology, all of whom could be linked to the constellation Centaurus: Kheiron (Chiron, Χείρων), the wise and educated Centaur who taught heroes like Achilles, Theseus, Iásōn, and Hēraklēs, who--although looked like the Centaurs of mount Pelion--was sired much earlier, by Kronos; Pholus, a friend of Hēraklēs, who died of the same poison that led Kheiron to give up his immortality; or Nessus (Νέσσος), the Centaur who carried Hēraklēs' third wife Deïaneira (Δῃάνειρα) over the river Evinos (Εύηνος), and was killed by Hēraklēs for attempting to abduct and rape her. Many of these, however, are linked to the constellation Sagittarius, not Centaurus, and as such, I'll get back to these when I reach this constellation. It seems only Centaurus may rightfully be linked to Centaurus.