(covers information from several alternate timelines)
Dom-jot was a game played with a ball and cue on a table with an irregular geometric coordination, similar to Terran billiards with certain elements of pinball. Rolling the terik into straight nines was considered an extremely skilled move in dom-jot.
In 2327, Jean-Luc Picard was stabbed through the heart by a Nausicaan during a quarrel over a game of dom-jot on Starbase Earhart. This incident led to Picard undergoing emergency surgery to receive an artificial heart. (TNG: "Tapestry")
In 2371, Benjamin Sisko learned from Jake's then-girlfriend, Mardah, that Jake was quite a dom-jot hustler. Later that year, Jake cancelled plans to play dom-jot with Nog, in favor of a date with Leanne at Deep Space 9's Klingon restaurant. (DS9: "The Abandoned", "Life Support")
In 2372, shortly before Nog departed for Starfleet Academy, Jake mused that they seemed to have spent most of the previous three years doing nothing, aside from playing dom-jot and watching Bajoran transports dock. (DS9: "Little Green Men")
While standing in Jake and Nog's customary spot above the Promenade and watching the incoming passengers, Jake was feeling lonely while Nog was away at the Academy. In defense of Nog's decision to join Starfleet, however, and despite his feelings, Jake told Quark that he was proud of what Nog was doing. Quark replied, "Remember that the next time you try to play dom-jot by yourself." (DS9: "Shattered Mirror")
The dom-jot table seen in "Tapestry" was designed by Production Illustrator Rick Sternbach who noted the playing surface as being made out of felt with rubber bumper strips around the outside of the table, which itself was held up by three "V"-shaped legs. (Star Trek: The Next Generation - The Continuing Mission, p.178)
The script for "The Maquis, Part II" includes a conversation between Jadzia Dax and Kira Nerys, in which Dax recalls that Curzon Dax once persuaded Benjamin Sisko and Calvin Hudson to play a game of dom-jot against two Zakdorn on Pelios Station. The Zakdorn won each game played over seven hours and Curzon revealed he had bet on their victory. This conversation was ultimately cut from the finished episode. 
Several dom-jot balls, described as "acrylic spheres about the size of a billiards ball, in quasi-transparent colors with a diameter of approximately two inches", were later sold off in the It's A Wrap! sale and auction for $565.55. 
According to the Pocket TOS novel The Shocks of Adversity, the game was invented by the Nalainger race, and is first encountered by the crew of the USS Enterprise (NCC-1701) while on a mission near the Goeg Domain in 2268, during their five-year mission.
Legends of the Ferengi reveals that Nog ran a dom-jot gambling ring during his time at Starfleet Academy.
Just prior to the Attrexian space station mission in the non-canon video game Star Trek: Elite Force II, Ensigns Jonathan Struhlem and Brian Bobowski can be overheard discussing dom-jot in the shuttlebay of the USS Enterprise-E. Following the death of Ensign Franklin in the previous mission, Struhlem states he would miss their regular dom-jot sessions, to which Bobowski replies that he never understood the game, and prefers poker instead.