A ball was an object, usually spherical, that could be used in sporting events or as a simple toy.
A 20th century sport, basketball used a similarly named ball. This sport was still played, at least informally, in 2154. (ENT: "The Forge")
The American sport of football, not to be confused with football, used an ellipsoid shaped ball, named after the sport. (ENT: "Fortunate Son")
Soccer was a sport that used a ball one would kick on the ground. (ENT: "Minefield"; TNG: "Lower Decks")
Racquetball was a sport that by the 24th century used small metal rackets and a small green, sometimes glowing, ball. (TNG: "Suddenly Human", "Rivals")
The sport of water polo also used a ball. Jonathan Archer had his water polo ball aboard Enterprise, though it rarely saw its intended use. (Star Trek: Enterprise)
In 2267 when Uhura had been altered by Nomad and was speaking in Swahili, Nurse Chapel had her say the line "The dog has a ball," a phrase that took her a few tries. She followed it up by stating that "the ball is blue." (TOS: "The Changeling")
Benjamin Sisko kept a baseball on his desk aboard Deep Space 9 and encouraged an interest in the game of the same name. (DS9: "Take Me Out to the Holosuite")