All of these words are closely related and have overlapping meanings. "leisure activity" is probably the broadest of these terms - a "leisure activity" is something you do when you are "at leisure" i.e. not working. "Entertainment" tends to be passive in nature. Theater, opera, symphony, ballet, movies, television are all "entertainment". "Recreation" tends to be more active in nature. Sports (tennis, skiing, swimming, etc.), boating, hunting, camping, hiking are recreational activities. "Amusement" is something that "amuses" you and is most commonly associated with things like amusement parks, video arcades, miniature golf, etc. I'm sure these definitions could be improved upon but that is the best I can do "off the top of my head".