Chess is a two-player strategic board game that Professor X and Magneto often play when they are together. It symbolizes their "hostile-friend" relationship. The game is references several times in the franchise.
While visiting him in his plastic prison, Charles was playing chess with Magneto. As Charles leaves, Erik knocks his king over, indicating that Charles won the game.
During the War on Alcatraz, Magneto applied the principles of chess into battle.
Near the end of the film, Magneto (having recently lost his powers) is in a park, where makes a metallic chess piece wobble. This indicates that his powers are returning.
Charles and Erik were playing chess on the steps of the Lincoln Memorial in Washington, D.C. They were later seen playing chess at the X-Mansion, while they were discussing Shaw starting World War III.
On their way to Paris, Charles and Erik settle their differences over a game of chess.