Utilitarianism is a moral theory generally considered to have been founded by Jeremy Bentham, a 19th century English philosopher and social reformer. It is centered around the concept of happiness, and seeks to promote it. The idea here is that all people seek happiness, and that it is the ultimate goal of all human beings to be happy. Therefore, according to classical utilitarianism, when a person wishes to act in an ethically sound manner he or she should strive to bring about the greatest possible amount of happiness for the greatest possible amount of people. This is known as the greatest happiness principle. Another, similar idea is that a person should always strive, if incapable of producing happiness, to reduce unhappiness.
To help better understand what John Stuart Mill's position on abortion would be let's try and better understand what Utilitarianism is. In the second chapter of Utilitarianism titled "What Utilitarianism Is," Mill states that "Utility, or the Greatest Happiness Principle, holds that actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness. By happiness is intended pleasure, and the absence of pain; by unhappiness, pain, and the privation of pleasure." This is a rather long quote but it does a good job of summarizing the viewpoints of John Mill. In this quote, Mill basically states that the rightness and wrongness of actions depend on how much pain or happiness you feel. Actions that receive more happiness will be seen as right and actions that receive more pain will be seen as wrong. Mill goes on to say that the amount of pleasure and pain depend on the quantity and quality of the specific pleasures and pains you feel. After all of this was stated and understood in class, it only makes sense that the Greatest Happiness Principle would give the great amount of pleasure to the greatest number of people.