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.
Von Neumann and Morgenstern proved the expected utility theorem, which says that when a rational agent chooses between different "gambles" (probability distributions over outcomes), the utility of such a gamble can always be seen as the expected utility of the gamble's outcome.
Humans, of course, are a different story.