The game is basically a version of "Rock, Paper, Scissors," except "scissors" is replaced with a stick of butter.
Role in episode
Patrick tries to make his own game, which leads him to create a game called "Rock, Paper, Butter." He tries playing it with Squidward, but Patrick gives up on the game when his butter starts melting. Squidward then tells Patrick that he "butter invent a different game," and Patrick ends up eating the butter.