Psychological egoism (see also nihilistic psychological egoism and absurd psychological egoism) is the position that all humans act only in self-interest. On the surface this seems like a plausible statement. As always, however, let's go through the process.
Definitions are in order. First off, we must understand what we mean by "self-interest." The first major problem that can arise when discussing psychological egoism is confusing self-interest and selfishness. Self-interest is acting to benefit myself. Selfishness is acting to benefit myself and harming others in the process. All squares are rectangles, as it were. Obviously not all acts are selfish (I take it here that it is possible to act without harming someone; if you disagree, feel free to write it up). The question is still in the air about self-interest.
Now then, what does the psychological egoist mean when he says, "All human acts are done in self-interest."? He cannot mean literal physical health. The classic example is a person who throws himself on a grenade to save his companions. The egoist perhaps here replies, "Some people value honor and how others see them more than their own life, and so killing yourself to save others is really only self-interest." That doesn't seem terribly convincing if you think about it. Does a person really have time to think about honor and self-image in the instance of a grenade? Split second decisions leave no room for imagining glory before acting.
"Fine," says the egoist, "but you can't despute that all actions are based on desire. No one does anything unless they want to, even if the desire stems only from an attempt to avoid consequences." This seems reasonable enough. What is not reasonable, however, is it what it presuposes: that all desires are self-interested. I see no reason to think that my morals must be based on my desires; rather, most of us spend our lives trying to curb our desires to fit our morals. If I help someone because I want to, does that make the act self-interested? Wanting to be charitable can be an act outside of self-interest. Yes, the egoist is right when they reply that some people are charitable so that they will be seen as good people when they aren't or for some other ulterior motive, but that does not necessitate that all people are charitable for those reasons. Once you start doing your own digging, the psychological egoist is left with little solid ground to stand on.