Hoshi Sato with a ball used in the sport of basketball
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 used an ellipsoid shaped ball, which was 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"; PIC: "The Impossible Box", "Et in Arcadia Ego, Part 1", "Et in Arcadia Ego, Part 2")
The sport of water polo also used a ball. Jonathan Archer had his water polo ball aboard Enterprise NX-01, though it rarely saw its intended use. (ENT: "The Catwalk")
In June 2151, Captain Jonathan Archer told Thy'lek Shran that "a man in Canton, Ohio" had once rolled a ball of string six meters in diameter. (ENT: "The Andorian Incident")
In 2267, when Nyota Uhura's memories had been purged by Nomad and she was speaking in Swahili, Nurse Christine Chapel had her say in English 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")
The Edo also had balls as toys for the younger individuals. Three of them played with the ball when Wesley Crusher joined them in 2364. (TNG: "Justice")
Racquetball was a sport that by the 24th century used small metal rackets and a small green, sometimes glowing, ball. (TNG: "Suddenly Human"; DS9: "Rivals")
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")