The recreation facilities on Starfleet vessels were typically equipped with several board games of Terran or alien origin. These games were quite popular among Starfleet crews as an off-duty pastime. There were two-dimensional as well as three-dimensional board games. 2D games were played on a single, flat game board, whereas 3D games usually featured several flat game boards on different planes.
2D board games
The recreation room on the Enterprise NX-01 was equipped with a standard Terran chess set. (ENT: "Observer Effect") Ensign Jean-Luc Picard also kept a chess set with large, frosted glass pieces in his quarters at Starbase Earhart. (TNG: "Tapestry")
3D board games
A very popular 3D board game in Starfleet was three-dimensional chess. 3D chess sets were present in the recreational facilities aboard the USS Enterprise, the USS Defiant, and the USS Enterprise-D, here most notably in Ten Forward. (TOS: "Where No Man Has Gone Before"; ENT: "In a Mirror, Darkly"; TNG: "Q Who")
The recreation room on the Constitution-class Enterprise was also equipped with a 3D board game consisting of four vertical acrylic rods in a quadratic arrangement, where each rod supported three 2x2 game boards which closely resembled the moving boards on a 3D chess set, so that there were twelve boards with a total of forty-eight squares (twenty-four white and twenty-four transparent). The game was played with red and black checkers pieces, eight of each color. (TOS: "The Naked Time"; DS9: "Trials and Tribble-ations")
Ten forward on the Enterprise-D was equipped with a three-leveled board game. The game set consisted of two chess-like 7x7 checkered boards on the upper two levels, and a lower board of the same overall size, but closely resembling a Nine Men Morris board. The three levels were arranged stair-like. The game was played with silver and gold game pieces which were all alike. (TNG: "Pen Pals")