Movement of pieces is similar to that of traditional chess. The main difference is that in the course of a move, pieces may move up or down any number of levels.
Commander Spock was an exceptional chess player, his game was consistently logical. However, he often had a difficult time predicting or effectively responding to the unexpected moves made by his frequent opponents, Captain Kirk and Doctor McCoy. Spock introduced chess to young passenger Charles Evans, who disliked the involved nature of the game. After losing he destroyed Spock's chess pieces. Spock also played a game of chess against the Kelvan expedition leader, Rojan. He observed during their match that Rojan's game was "off," which suggests that it was not the first time they had played each other. (TOS: "Where No Man Has Gone Before", "Charlie X", "By Any Other Name")
Spock also enjoyed playing chess against a rival logical mind, that of the Enterprise computer. In 2266 he detected programming errors in the computer's databanks because of faulty chess moves made by the computer. He later introduced the tampering and unreliability of the computer's records as defense evidence in the court-martial of Captain Kirk. (TOS: "Court Martial")
When the Enterprise was apparently helpless against the alien Balok's threats to destroy them, Spock compared the situation to chess, suggesting that they were checkmated and the game was over. However, a comment from McCoy led Kirk to reject Spock's chess analogy and try playing poker with the alien instead. (TOS: "The Corbomite Maneuver")
Kirk and his senior officers used a chess-based code phrase as transporter clearance in 2268, when Garth of Izar planned to escape from the Elba II insane asylum. Chief engineer Scott declined to beam Garth, disguised as Kirk, to the Enterprise, when after he challenged Garth with the code phrase "queen to queen's level 3" and Garth could not respond with the correct pass phrase (which we learn to be "queen to king's level 1" when Mr. Spock calls for transport back to Enterprise) (TOS: "Whom Gods Destroy")
Counselor Deanna Troi managed to beat Lieutenant Commander Data at a game of 3-D chess in Ten Forward, prompting Data to honor their bet to make Troi a Samarian sunset in the "traditional style". (TNG: "Conundrum")
Commander William T. Riker defeated both Ferengi Doctor Farek and a Ferengi guard at a presumably-alien variant of 3-D Chess; the former defeat again in Ten-Forward, and the latter while a captive of the Ferengi along with both Deanna and Lwaxana Troi. (TNG: "Ménage à Troi")
Commander Benjamin Sisko, also a fan of the game, kept a three-dimensional chess set in his quarters aboard Deep Space 9. (DS9: "Move Along Home", "The Nagus", "The Maquis, Part I", "Statistical Probabilities")
The call sheet from the episode "The Schizoid Man", dated on 4 November 1988, is listing the three-dimensional chess as one of the required items from the art department in the special instructions section. Here it is named "Okuda chess set".
The novelization of Star Trek makes a quick - and foreshadowing - reference; as the two fight their way through the Narada (largely a fistfight rather than the gun battle seen in the film), Kirk, marveling at Spock's highly effective use of Suus Mahna (thinking to himself, "he even fights logically"), concluded, correctly, that the Vulcan must play a "mean game" of 3-D chess.