The game is played on a court, where the players have to serve and return the ball to each other using rackets. When a player lets the ball go out of play or hit the net the opposite player scores.
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. (DS9: "Melora")
The Bajoran criminal Ibudan was an avid tennis player, having scheduled practices in the holodeck between stardate 46383-46385, according to his personal calendar that Odo was reviewing. (DS9: "A Man Alone")
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")
Background information Edit
There seem to be a number of different tennis technologies in the 24th century. For example, in "Distant Voices" the rackets that Dr. Bashir and Elim Garak play with use strings, while Guinan's racket seen in "Suspicions" appears more futuristic and seems to be totally made out of metal.  Likewise, the tennis balls that appear in Bashir's dream are much more traditional looking than those that appear in "Future's End".