Answer
Playing chess and backgammon is absolutely forbidden, (with or without betting).
As for playing other games, then playing them with the condition of betting (gambling) is forbidden (Haram), and one should avoid playing these games (known as games of gambling), dominoes, billiard, and card games even without betting (Obligatory Precaution).
Any other game which is not considered a game of gambling, and no betting is involved, is permissible to play.