As I sometimes put it: When you boil it down, pleasure is pleasurable, and pain is painful. Every other action we take should derive from that.!
This is the central utilitarian guideline, and also its essential flaw. To whom does this doctrine apply? If we apply it to individuals, well, it just so happens that I'd probably feel pretty good about shooting Bill O'Reilly right now, but I'll bet you he wouldn't. Conversely, if we apply it to societies, then individuals on their own become worthless in the name of the greater good. To whom does this doctrine apply, for it to succeed in its intent?
The conflict between "do no harm" and "seek pleasure but avoid pain" rests on the fact that people's own definitions of pain and pleasure are often mutually exclusive from one another. (That's why I mentioned Paradise with a capital P in my little essay earlier; it implies a hypothetical state where no one's paradise treads on anyone else's.)
So how do we get around this minor problem, Mr. Guru of Time? Is the flaw with utilitarianism one that can be solved?
Easily! Just pretend you're the government and deny any evidence that there's a flaw at all!
...
Okay, seriously. The problem here comes with being shortsighted, again. In the end, if everybody gives weight to the pleasures of others, each person's individual pleasure gets even higher.
But then person A says: Wait! But I can get +1 Pleasure if I give person B +2 Pain! What have I got to lose? So, person A is at +1 overall happiness, person B at -2.
Then person B says: You know what? I can do that too! He even deserves it, the bastard. So, he goes right back and gives person A +2 pain, gaining +1 pleasure.
In the end, persons A and B are both at -1 overall happiness, and both probably wish that in retrospect this incident had never happened.
This is an oversimplification, but the message is valid: Generally, when you gain some pleasure at the expense of another, you gain less value than they lose, harming the sum happiness of humanity. Your behavior then encourages others to do the same back, and you end up becoming a victim as well.
Thus, if everyone puts themself first in the short-term, everyone ends up worse off in the long-term.