A bowling alley was a place where the sport of bowling was played.
In 2266, Kevin Riley, under the influence of polywater intoxication, "ordered" a formal dance to be held in the bowling alley on the USS Enterprise. (TOS: "The Naked Time")
Deep Space 9 also had a bowling alley on its Promenade. Its address was 01-854. (Star Trek: Deep Space Nine set decoration)
The bowling alley Easter egg
The bowling alley on Kirk's Enterprise was located on deck 21 in the Star Trek Blueprints. It was depicted in an easter egg in the aborted PC game Star Trek: Secret of Vulcan Fury.