Friday the 13th is a game developed by Atlus and released by LJN in 1989 based on the popular ongoing series of the same name.
According to the manual, the undead killer, Jason, has returned to Crystal Lake, and is wrecking havoc on a nearby summer camp. It is up to the camp counselers to stop Jason's rampage while fighting off zombies, bats, wolves, and other foes.
The game's artwork displays a picture of Jason from the most-current movie at the time, Friday the 13th Part VII: The New Blood, however Jason's in-game appearance is based off the killer in Friday the 13th Part V: A New Beginning, which interesting enough wasn't even Jason, but an imposter. Since the ending of Friday the 13th Part VII: The New Blood shows Jason's still alive, this game logically takes place after the events of that movie (however, in the movie, Jason lost his trademark hockey mask during the climax).
References to the movies
While not based off any film in the Friday the 13th series in particular, it pays homage to many of these films in different ways.
The intro animation is based off artwork for Friday the 13th: The Final Chapter.
Jason's hidden cabins in the woods are a reference to his hidden dwelling in Friday the 13th Part 2.
Also from Friday the 13th Part 2, Jason had room dedicated as a shrine to his deceased mother, Mrs. Voorhees (the main antagonist of Friday the 13th). Just like in the movie, her decapitated head is prominently displayed in the center (which is a miniboss in this game). One of the items that you can access after defeating her, a sweater belonging to her, is also based on how it was used in the movie by Ginny. In the game, the sweater reduces damage by 50%, which is a reference to how Jason hesitating to attack Ginny while wearing it, mistaking her for his mother.