This thread is an off-shoot from 'Searching for God'.
I probably don't have to tell anyone here, but I will anyway, that utilitarianism is the ethical system that judges the rightness or wrongness of acts by whether they tend to increase happiness or reduce unhappiness (right), or the opposite (wrong). There are other sytems, such as deontology, which says that morality should be based on a set of fixed rules such as the ten commandments, rather than on the consequences of actions, as in utilitarianism. It seems to me that all other ethical systems, under investigation, collapse into utilitarianism. If you believe that morality is based on the 10 commandments, it is reasonable to ask why anyone should obey them, and the usual answer will boil down to "because it leads to greater happiness", so deontologists turn out to be utilitarians after all. Utilitarianism does not need to be questioned in that way: "Why should we seek to maximise happiness?" is a foolish question, the only sensible answer to which is "because it's happiness!". We all do try to maximise happiness for ourselves anyway, and it only takes a modicum of imagination to realise that, since everyone is doing it for themselves, a maximally happy society will be one in which everyone acts in such a way as to maximise general happiness.
There are two major types of utilitarianism: act and rule. Act utilitarianism applies the principle of maximising happiness and minimising misery to individual acts, which I think is impractical, whereas rule utilitarianism derives general rules from the principle, and judges acts according to the rules, though in exceptional circumstances it might be right to break a rule. Thus murder is normally wrong, but may in exceptional circumstances be justified, such as killing Hitler in 1938. I am a rule-utilitarian.