Mark Rosewater is the head designer for Magic: the Gathering. He's been writing a weekly column on M:tG and game design for the last decade or so. In his latest column, What Is a Game?, he sets out his definition of a "game":
MARK'S DEFINITION OF A GAME
A game is a thing with a goal (or goals), restrictions, agency, and a lack of real-world relevance.
Let me walk through each part of this definition.
A goal (or goals)
There needs to be a point to a game. What exactly are the players trying to do? If there's a way to win the game, how do you win? If there's a way it ends, how does it end? Players in a game need motivation, they need something to direct their actions. That comes from having a goal or goals. Now the goals can be active (defeat the enemy) or passive (don't die), but they must give the players some idea of what they're supposed to be doing.
Games are about obstacles. The players have a goal, but something keeps them from simply accomplishing it. A game needs to have some challenge to it because the fun of a game comes from figuring out how to overcome those challenges.
A game needs to have decisions, and those decisions must matter. Having a choice where the proper way to play is always making the same choice is not really a decision, and as such is not giving the player agency. Player involvement in the game and its outcome is core to the experience of a game.
Lacks real-world relevance
A game is something that you opt into doing because you want the experience of playing it. Labeling every obstacle you run into in life, a game quickly robs the term of any meaning. We use the expression "play a game" because it's an activity we opt into for some gain (usually entertainment and/or education, but there are many reasons one can chose to play).In the rest of the article, Rosewater goes through what happens if you have three of the elements, but not the fourth. It's a creative way of examining the issue, and is very revealing.
Perhaps the most interesting section is agency. Rosewater asserts that what is important is that you believe you have agency, not if you actually have it in reality. For example, he says that Tic-Tac-Toe is a game if you believe you can win. Once you realize that you can't win, it ceases being a game for you!
In any case, it's one of Rosewater's best articles, and I strongly encourage everyone to read it.