The game was played on a court, on which the players had to serve and return the ball to each other using rackets. When a player allowed the ball go out of play or hit the net, the opposite player scored.
In 2266, Captain James T. Kirk privately expressed, to Doctor Leonard McCoy, that he wished to be on a long, relaxing sea voyage and mentioned that, among the things there would not be, there wouldn't be too much deck tennis. (TOS: "Balance of Terror")
Julian Bashir was a tennis fan, and didn't follow baseball or soccer. (DS9: "Past Tense, Part II") He even considered becoming a professional tennis player, before deciding on a career in medicine. He believed he was good at it until his first opponent served first and knocked the ball past him. When the computer confirmed the move was good, he knew he was in trouble. (DS9: "Melora")
The Bajoran criminal Ibudan was an avid tennis player, having scheduled practices in the holodeck between stardates 46383-46385, according to his personal calendar that Odo was reviewing. (DS9: "A Man Alone")
In 2369, Guinan claimed to have played tennis with Geordi La Forge, who she said had beaten her in straight sets, causing her to develop tennis elbow. Later, while Beverly Crusher was telling her about Doctor Reyga's death, she recalled that La Forge "kept hitting to my backhand at the net". Crusher was surprised by the El-Aurian's seeming lack of concern over two deaths, preferring to focus on her game. Later, Guinan admitted that she had never played tennis before and that the whole thing was a ruse to get a concerned Beverly to open up to her. (TNG: "Suspicions")
Kathryn Janeway was a tennis player in her youth. At the age of twelve, she once walked home in a thunderstorm over a distance of more than seven kilometers because she had lost a match. (VOY: "Deadlock") Later, she played on her high school tennis team. In 2373, she decided to take up tennis again, after giving it up in 2354. Tuvok walked in on her, in her ready room, as she was practicing her serve, and was nearly hit by the ball. She then confessed that she was a bit rusty, after losing her first match in straight sets in a novice tournament on the holodeck. Tuvok suggested that she should "maintain eye contact with the ball at the apex of its trajectory", so that her "serve would be more effective". When asked by Janeway if he was a tennis player, he logically stated that it was "simple physics". (VOY: "Future's End")