In 2154, Ensign Travis Mayweather and Lieutenant Malcolm Reed played chess while inhabited by Organians. Lieutenant Reed told Ensign Mayweather that he was the best player of chess aboard the NX-class starship Enterprise. (ENT: "Observer Effect")
By 2265, a three-dimensional variant had been devised and was available to the crew of the Constitution-class USS Enterprise. James T. Kirk and Spock frequently enjoyed playing it together. (TOS: "Where No Man Has Gone Before", "Charlie X", "Court Martial", "By Any Other Name", "Whom Gods Destroy")
In 2266, during a confrontation with the First Federation vessel Fesarius, Spock likened the situation to chess in which one is lost when overpowered. However, Kirk noted that it's rather like poker, a game in which one can overcome a losing situation by bluffing. (TOS: "The Corbomite Maneuver")
Despite the game's inherent complexity, some individuals have been dismissive of it. When Harry Kim referred to Kal-toh as "Vulcan chess", Tuvok claimed that "Kal-toh is to chess as chess is to tic-tac-toe". (VOY: "Alter Ego")
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Background information Edit
Three chess sets "like the one featured in the Star Trek: Enterprise episode 'Observer Effect'" were sold during the It's A Wrap! sale and auction on eBay, and two three-dimensional chess sets were sold in the Christie's auction 40 Years of Star Trek: The Collection. 
In a revised draft script for TOS: "The Galileo Seven", a game of chess, played in Captain Kirk's quarters, was featured at the start of the episode's teaser. In a memo to Gene Roddenberry (dated 23 August 1966), Robert Justman suggested omitting the chess game and instead having the entire teaser set on the Enterprise's bridge.
An unfilmed line from TNG: "Coming of Age" suggests that the Mordock Strategy was a strategy in Benzite Chess. The script of "Empok Nor" described the Cardassian game of Kotra as being "a cross between chess and Stratego".
The chess nomenclature used for the computer's lines in TNG: "Evolution", English Descriptive Notation, has become highly uncommon in the early 21st century, in favor of the far more efficient Algebraic Notation method, which was developed during the early decades of the 20th century.