Expected utility is the expected value in terms of the utility produced by an action. It is the sum of the utility of each of its possible consequences, individually weighted by their respective probability of occurrence.
A rational decision maker will, when presented with a choice, take the action with the greatest expected utility. Von Neumann and Morgenstern provided [http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem#The_axioms| 4 basic axioms of rationality]. They also proved the expected utility theorem, which states a rational agent ought to have preferences that maximize his total utility. Humans often deviate from rationality due to inconsistent preferences or the existence of cognitive biases.