(covers information from several alternate timelines)
A birthday party was an event held in the honor of an individual, celebrating the anniversary of the day on which they were born.
Commander Riker celebrated his birthday on the USS Enterprise-D with a birthday party. Lieutenant commander Data found the tradition of a birthday party interesting, as the day of their birth was something that Humans could not possibly remember. (TNG: "Future Imperfect")
Several months later, Riker wanted to give Worf a surprise party for his birthday, but Deanna Troi talked him out of it. In a number of alternate quantum realities, however, the party did in fact take place. (TNG: "Parallels")
For Kes' second birthday in 2371, the USS Voyager crew threw a surprise party for her. They explained the event (Kes found it exhilarating) and instructed her on the Earth tradition of making a wish and blowing out candles. Neelix made a seven layer Jimbalian fudge cake with l'maki nut icing and punch. Tom Paris replicated a locket as a gift. (VOY: "Twisted")
Keiko O'Brien prepared a surprise birthday party for her husband Miles in 2373, but during the party she actually was possessed by a pah-wraith. For the party she prepared q'parol, which was a big success. (DS9: "The Assignment")