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]. Under these axioms, they proved the expected utility theorem, which says that exists a utility function defined on possible outcomes and every decision maker's preferences are characterized by maximizing the expected value of the function. The expected utility hypothesis holds that these axioms truly model the notion of a rational agent. Either way, humans often deviate from ratinality by committing cognitive biases.