A finite game is a game that has fixed rules and boundaries, that is played for the purpose of winning and thereby ending the game.
An infinite game has no fixed rules or boundaries. In an infinite game you play with the boundaries and the purpose is to continue the game. Finite players are serious; infinite games are playful.
Finite players try to control the game, predict everything that will happen, and set the outcome in advance. They are serious and determined about getting that outcome. They try to fix the future based on the past.