(covers information from several alternate timelines)
A baseball was a spherical object, used in the game it was named for. Measuring approximately 0.22 meters in circumference, and 140 grams in weight, it was made of a lightweight core with a leather outer, stitched together by 108 stitches of red cotton.
In 2369, Benjamin Sisko received a baseball from an alien impersonating legendary baseball player Buck Bokai. He kept it on a stand in his office on Deep Space 9 and often tossed it between his hands during tense moments. He had previously kept a baseball with glove in his quarters. (DS9: "If Wishes Were Horses", "The Nagus", "The Homecoming", "The Circle")
During the initial phase of the Dominion War, after the Dominion temporarily gained control of Deep Space 9, the baseball was left behind, serving as a reminder to Dukat, who had taken command of the station, that Sisko intended to return. (DS9: "Call to Arms") Sisko was given the ball back by Dukat following the latter's descent into madness, and the recapture of the station by Starfleet in Operation Return. (DS9: "Sacrifice of Angels")
In 2374, following the sealing of the Bajoran wormhole by the Pah-wraiths, Sisko returned to Earth on leave. Kira Nerys noted that he took the baseball with him – a sign that Sisko was not sure that he would be returning. (DS9: "Tears of the Prophets")
The ball also helped Sisko locate the Orb of the Emissary on the planet Tyree three months later. In a vision sent to him by the Prophets, the ball rested at the location where Sisko was digging. When Ezri Dax took the ball from him on the surface, and threw it away, Sisko realized that the Orb was located where it fell. (DS9: "Image in the Sand", "Shadows and Symbols")
Sisko was given another baseball shortly thereafter, signed by the members of his senior staff in the Niners baseball team, after their game against the Logicians in 2375. (DS9: "Take Me Out to the Holosuite")
The baseball marked a symbolic passing of power by the end of that year – following Sisko's ascension to be with the Prophets, and Kira Nerys's promotion to station commander at the close of 2375, she picked up the ball for the first time. However, the fact that Sisko left the baseball behind also was likely a sign that Sisko would return someday. (DS9: "What You Leave Behind")