The basic idea of the utility theory is very simple: agent preferences over possible results can be captured by a function that compares these results with a real number; the higher the number, the more he likes this result. A function is called a useful function.
For example, you can say that my utility for storing various elements:
u(apple) = 10
u(orange) = 12
u(basketball) = 4
u(macbookpro) = 45
Economists (usually) consider people agents that maximize utility. That is, we always try to maximize our internal utility function.Once you have these numbers, you can mix them with probabilities and talk about expected utilities, optimal strategies, discounts for future rewards, and many other interesting things.
If you want to know more, take a game theory book or read the first chapter of this book of agents .