What is Utilitarianism?

On a simple level, utilitarianism is the belief that the morally right thing to do in any situation is to maximise the greatest happiness for the greatest number of people.
A good example of this idea in practice is the organ donor problem. There are 4 patients who are in a hospital waiting room waiting for a life-saving transplant (needing a heart, liver, kidney, and lung, respectively). There are no organs available, and all 4 are set to die. A healthy man comes in for a check-up. The Doctor (a true utilitarian), knocks the healthy man out, takes his organs, and transplants them into the 4 waiting: saving their lives. In a utilitarian calculus, this the morally right thing to do because it saves 4 lives for the cost of 1; it maximises happiness for the greatest number of people.