What is utilitarianism?

Utilitarianism is one of the major moral theories along with deontology and virtue ethics.

According to utilitarianism, the morally best action is the action that maximises utility (or good) to the greatest extent.
So in utilitarianism, it is the outcome of an action that determines its moral worth and not its intention.

Imagine the following scenario: In order to save five people's lives, you will need to kill two people. 
Now the utilitarian will decide that (because 5 > 2), more good or utility will be established by killing two people and thereby saving five others. So it will be morally right, according to utilitarianism, to let five people live and kill two.

Can you think of implications of this moral theory? 
Can you think of a scenario where killing the two people might not be the morally best action?
Can you think why this moral theory might be appealing to philosophers?