What is Pascal's Wager?

Pascal's Wager originates with French thinker, mathematician, and scientist Blaise Pascal (1623–1662). Before detailing the argument itself, it is important to note that the Wager is actually a very small part of a much larger body of work. This fact should be recognized before proceeding any further.
What is particularly noteworthy about the Wager is where it diverges from most theistic arguments. If we were to look at the Ontological Argument or the Cosmological Argument, for example, we'd be looking at a logical proof for the existence of God—if we accept the claims and the arguments made, then we must accept God's existence as a rational conclusion. Pascal's wager, on the other hand, doesn't try to prove God's existence at all—rather, it tries to defend the belief in God, and moreover, it does so on pragmatic grounds, as a kind of optimal selection among various choices.
Ultimately, for Pascal, there are two sets of general possibilities: on the one hand, a person can either believe or disbelieve in God, and on the other, God will in the end either exist or not exist. Pascal's Wager weighs both sets of outcomes against each other: If the nonbeliever is correct, and there is no God, then they were right and the believer wrong. However, if the believer is correct, and God has existed all along, then the stakes get much much higher, for believer and nonbeliever alike. It's a kind of odds game, and for Pascal, the best choice is the one that gives the best outcome—in this case, belief in God.