Utilitarianism is a theory in philosophy about right and wrong actions. It says that the morally best action is the one that makes the most overall happiness or "utility" (usefulness). This is not limited to the happiness caused by a single action but also includes the happiness of all people involved and all future consequences.
The theory essentially states that an action is justifiable if it brings the most amount of happiness for the greatest number of people.
The theory was made popular by 18th and 19th century British philosophers like Francis Hutcheson, Jeremy Bentham, and John Stuart Mill, but the idea goes all the way back to ancient times.
Bentham wrote about this idea with the words "The greatest good for the greatest number", but did not use the word utilitarianism. It was Mill, a follower of Bentham's ideas, who named the idea.

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