The Mesoamerican ball game is the oldest known team sport in the world, and undoubtedly the one with the most complex and significant history. It dates back to at least 1500 BCE, when the earliest known ball court was built at Paso de la Amada, Mexico, perhaps almost a millenium before the establishment of the Greek Olympic Games, and is still played, in a modified form, today. It seems to bave been played across a huge swath of the Americas, from Paraguay to the Southwest United States, and into the Caribbean. We know from surviving artistic and textual evidence that it was of tremendous religious and social importance to both the Aztecs, who called it tlachtli, and the Mayans, who called it pok-ta-pok, and it's reasonable to assume it had a similar role in many of the other cultures where it was played.
The rules of play varied to a certain extent, as is to be expected of a sport played over three and a half millenia. It was played by two teams of between two and six players, who attempted to manipulate a solid elastic rubber ball past a goal line, or, later, into a suspended stone hoop, without using their hands or feet, and without allowing it to touch the ground. It was played on a I-shaped (imagine an I in a serif typeface here) or occasionally oval court, sometimes with banked sides; the size of the courts varied, probably to accomodate different-sized teams, since many archaeological sites have several different courts of different sizes. The courts varied in size from roughly the size of a tennis court to roughly that of an American football field. The game was played with protective equipment, which varied from culture to culture, but usually included a padded yoke worn around the waist as a striking surface for the ball.
The one thing modern Westerners are most likely to know about the game is that the losers were made into human sacrifices. This did take place, though not, unsurprisingly, after every game, and probably not in every era and place the game was played. The Spanish Conquistadors, for instance, encountered the game and reported that it was played purely as a form of recreation, and while they often didn't know their asses from a hole in the ground so far as the cultures they encountered were concerened, it would be utterly out of character for them to miss an opportunity to mention and condemn any practice related to sacrifices, and they seem to have been largely amused by the game.
The sacrifice of ball-players was intimately related to the celestial cycle of the sun and moon for both the Mayans and Aztecs, as was the game itself. One of the most important episodes in the Popol Vuh mentions two sets of important Mayan gods going down into the Underworld to contest with Lords One and Seven Death, the gods of the underworld, and afterwards being killed and transformed into celestial bodies. The sacrifice of losing teams in the ball game was a reaffirmation of this for the Mayans, and an aspect of a compact with the Underworld which allowed the sun and moon to rise every day so long as the sacrifices were made.
The Mesoamerican ball game, in terms of the vast range of time and space it was played in, and the immense importance it had for the people who played it, can honestly be considered the most important sport in human history. The impulse to play competitive sports seems to be one of the most primal, and heavily valued, in human nature.