This is a much debated topic, and I'm sure everyone has their own definition, but this is mine:

Game

noun - The decision space created when a group of individuals agree to abide by a particular set of rules.

Discussion

At this point you are probably astounded at the succinctness and accuracy of this definition, but if you are not entirely convinced there is more elaboration below.

Agreement

Inherent in this definition is the idea of agreement. People must agree on what the rules to the game are, and all agree to follow them during the duration of the game. If someone breaks this basic contract we give it a telling name, "cheating". Quitting the game (when not specified by the rules) consists simply of breaking this contract.

The rules are not necessarily the same for all players. This is demonstrated by games like Letters from Whitechapel where each side is playing by almost entirely different rules, yet which depend on each other to make the game work. Other games are less severely asymmetrical where all players share a core set of rules but still have distinct rules for different "characters" or "factions", like Chaos in the Old World. But in these cases all the players still agree that each player is playing by their own set of rules to whatever degree. The agreement is still necessary to be playing a game.

Decision Space

The decision space is the set of all possible actions that can be taken in a game. Every point in this space is a game state, some configuration or relationship of all the elements of the game. Making decisions in a game constitutes taking a particular path through this space from game state to game state. Actions taken constrain or expand the possible choices that can be taken later, by that player or other players. This is the space that the players explore over the course of the game, and players are usually trying to steer all of the paths in the current game towards some favorable end state, however that is defined (but not always!)

I used to think games like Candyland weren't games because the players don't actually make any decisions and ultimately have no real effect on the outcome. But now I see this as kind of the trivial case of a decision space, the Null space (to take an analogy from Linear Algebra). Even though no decisions are made, the players are still agreeing to abide by the rules (draw cards and move their pieces). It is like a zero of decision spaces. Zero is still a number, and distinct from the absence of something (as was settled in an unforgettable rules dispute during a game of Cutthroat Caverns at PDXAGE this year!) So I grudgingly accept Candyland as a game now. Barely.

What is not in the definition

A definition is as important in what it excludes as what it includes. This is the key to a good definition. Including everything or excluding everything is easy. Including just the right things while excluding just the wrong things is the hallmark of a good definition.

You may notice some glaring omissions in this definition. For instance, nowhere do I say that someone must win the game, or must have a goal, or even that it has to end! This is intentional, because there are legitimate games that never actually end within the game. Maybe people agree to stop playing for awhile, but this is a decision that exists outside of the game itself. It is a metadecision. Games like D&D are like this. Maybe you die, but you can always make a new character and continue. Maybe you finish the current adventure... but there is always another one possible. The only thing that causes a D&D game to end is the limits of humanity, not of the game itself.

Also, there is a whole class of games that do not have an ending. Check out the fascinating treatise on this subject called Finite and Infinite Games.

Et Cetera

This is one of my favorite philosophical quandaries, and I in no way consider it to be solved. I may change it in the future, and I am open to any and all alternatives. Please let me know your definition, and you may just convince me.