Short answers, pulled from the story.
The October Horse, or Equus October, was an annual animal sacrifice to Mars held on October 15 in the Campus Martius. Two-horse chariot races were run, and the right-hand horse of the winning team was killed with a spear. Its head and tail were then used in separate subsequent ceremonies.
The sacrifice coincided with the end of Rome's agricultural and military campaigning season. Verrius Flaccus recorded that it was performed ob frugum eventum, meaning either in thanks for the completed harvest or for the sake of the next harvest, since winter wheat was sown in autumn.
The tail was carried at speed to the Regia so blood could drip onto Rome's sacred hearth, where it was preserved for the Parilia festival on April 21. The head, adorned with bread pendants, was contested in a neighborhood fight between residents of the Subura and the Sacra Via; the winner displayed it on either the Tower of the Mamilii or the wall of the Regia.
The earliest reference is from Timaeus in the 3rd century BC. The latest known record appears in the Calendar of Philocalus from 354 AD, by which time Christianity had become the dominant religion of the Roman Empire. Ancient references span more than six centuries.
Georges Dumézil argued that both rites preserved traces of a shared Indo-European tradition. In both the Roman and Vedic cases, the victim was the right-hand horse of a chariot team. In the ashvamedha, the horse's head signified spiritual energy and the tail represented livestock fertility, paralleling the Roman use of those same body parts.
Ancient scholars noted the paradox: the Roman military relied on infantry, and the Roman cavalry was formed primarily from allies by the late Republic. Polybius specified that the victim was nonetheless a war horse. Mars was also associated with horses at his Equirria festivals and the equestrian Troy Game, and the horse's potency was thought to intensify through the exertions of the race.