Finite games follow a fixed set of rules that can not be altered during the play of the game. The purpose of a finite game is to win. Chess and football are two typical examples of finite games.
In stead, the purpose of an infinite game is to continue playing. Infinite games transcend time and invite anyone who is willing to play to join in. Infinite games I used to play were keeping a ball up in the air as long as possible, or building tree houses.
Too often we approach life as a finite game: we believe success to be of a scarce nature; you must follow a fixed set of rules to get there and there can only be a limited number of winners. The world becomes much richer when we approach life as an infinite game: the rules become ours to (re)invent, and the goal is to continue the game together.