A birthday party is when most Humans celebrate them being born on the same day it happened.
Commander Riker celebrated his birthday on the Enterprise-D with a birthday party. (TNG: "Future Imperfect")
In 2370, Riker wanted to give Worf a surprise party for his birthday, but Troi talked him out of it. (TNG: "Parallels")
Ensign Ahni Jetal celebrated her birthday on the USS Voyager (VOY: "Latent Image")