Short answers, pulled from the story.
A centaur is a creature from Greek mythology with the upper body of a human and the lower body and legs of a horse. They were said to live in the mountains of Thessaly, particularly in Magnesia and Mount Pelion, and were described as wild and untamed. They appear in Roman mythology and medieval bestiaries as well.
The centaurs were usually said to be the offspring of Ixion and Nephele. Nephele was a cloud fashioned in the likeness of Hera, and Ixion was deceived into seducing her. A second version makes them descendants of Centaurus, a deformed man who mated with Magnesian mares; Centaurus was either the son of Ixion and Nephele or of the god Apollo and the nymph Stilbe.
The Centauromachy was a battle between the centaurs and the Lapiths. It was triggered when the centaurs attempted to carry off Hippodamia and the other Lapith women on the day of Hippodamia's wedding to Pirithous, king of the Lapithae. The hero Theseus intervened on the Lapiths' side, and the centaurs were driven off or destroyed.
The most common theory holds that centaurs arose from the reaction of non-riding cultures, such as those of the Minoan Aegean world, to mounted horsemen, who might appear as a single half-human, half-animal creature. Bernal Diaz del Castillo recorded that the Aztecs had the same misapprehension when they first encountered Spanish cavalrymen.
In Archaic Greek art, Professor Paul Baur identified three forms: Class A, with a human torso at the horse's withers (which became standard); Class B, with human legs joined to the hindquarters of a horse; and Class C, with human forelegs ending in hooves. Centaurs also appear in Roman art, including the Great Cameo of Constantine dated to around AD 314-16, and on Pictish carved stones in north-east Scotland from the 8th-9th centuries AD.
The Kinnaras are half-man, half-horse mythical creatures from Indian mythology, depicted in ancient texts, arts, and sculptures across India. They differ slightly from Greek centaurs: the Kinnaras are shown as a horse with the torso of a man where the horse's head would be. The Kalibangan cylinder seal, dated to around 2600-1900 BC from the Indus Valley civilization, also shows centaur-like figures in a battle scene.