The game is basically a version of "Tic-Tac-Toe," except this version is played on a 2 by 2 grid.
Role in the episode
In "Patrick! The Game," Patrick tries to make his own game which led him to creating a fast and easier version of "Tic-Tac-Toe" called "Tic-Tac." He started playing it with Squidward who kept winning because he went first every time. After many games, Patrick gives up on the game.
When making the game, Patrick was trying to make an easier and faster version of "Tic-Tac-Toe."