Egoism is a theory that holds that the only thing people do care about (psychological egoism) or the only thing that they should care about (ethical egoism) is themselves.
The psychological egoist holds that every act is selfish. If we look at any act and trace it back to its fundamental motive we will discover that the agent was ultimately moved by his own pleasure or happiness or interest in avoiding pain or unhappiness or some other self-regarding concern. The well-being of others is only considered as a means to an end - as a way for the agent to experience pleasure or happiness themselves. Even the quintessentially selfless act of a parent sacrificing himself for his children rests on his interest in avoiding the pain of having lost a child, not on any concern for the child independent of this pain.
One form of ethical egoism comes directly from psychological egoism. Applying the principle of 'ought' implies 'can' (and 'cannot' implies 'it is not the case that one ought'), if an agent cannot consider anything but his own pleasure or happiness, it is not the case that they ought to consider anything but their own happiness.
Desirism makes a claim that, at first glance, may sound like a form of egoism. Desirism holds that people seek to act so as to objectively satisfy the most and strongest of their own desires. This might be interpreted to mean that everybody is interested only in their own desires and nothing else.
This appearance comes from a failure to distinguish between two different ways to relate desires to those who have them: desires in the self versus desires of the self. Psychological egoism holds that the desires that motivate an agent's actions are all desires in the self, while ethical egoism holds that all actions should be motivated by such desires. Desirism, on the other hand, is built on the idea that all of an agent's desires are desires of the self, but they may be desires in a great many things.
To say that psychological egoism holds that all intentional action springs from desires in the self can be expressed as follows. For every desire that P, if we look at P we will discover a proposition that refers back to the agent. "I desire that I experience pleasure," or "I desire my happiness," or "I prefer a state in which I am experiencing no pain."
Desirism, on the other hand, is built on the view that there is no limit to the propositions that make up the potential objects of a person's desires. The possiblities for "desires that P" are as limitless as the possibilities "believes that P". "I desire that my children are safe," or "I hope that we can find a cure for AIDS" or "I want to make sure that the human race does not become extinct," are all things that an agent can desire. None of these propositions refer back to the agent making the desire. It rejects the thesis that all desires are desires in the self,
Saying that an agent's actions are motivated entirely by desires of the self means that it is an agent's own desires that pull the strings and push the levers that cause an agent's intentional actions. This is not only true, it is logically necessary. If one agent were to hook up a device that allowed him to control the body of another remotely, then those actions would no longer belong to the agent being controlled. They would belong to the controller. An intentional act belongs to the agent whose desires are motivating that action.
The agent who is deeply concerned about impoverished children not getting proper medical care is certainly acting on her own desires when she goes to impoverished parts of the world to care for children. If she did not care, then she would not act. However, it is odd at best to say that the person who cares so much about the plight of impoverished children is "selfish".
Egoists often argue for their position by blurring this distinction. They will begin by claiming that all actions are selfish - motivated by desires in the self. When challenged with stories such as the possibility of the doctor mentioned above they retreat to arguments that defend the claim that all actions are motivated by desires of the self. When the opponent gives up, they will claim to have successfully defended the claim that all of an agent's interests are desires in the self
In addition to the considerations raised above, there are other problems with the idea that all agents are motivated by their own pain or pleasure.
It is an inefficient way to build a system. For a system that seeks food, it would be much simpler to design a system that can detect food and respond with behavior that aims at acquiring food than to design a system that seeks pleasure, somehow knows that acquiring food is pleasurable, and then seeks food, not for its own sake, but as a way of obtaining pleasure.
When we look at animals, we see the strangeness in the thesis that they seek only pleasure or happiness. In order to save the gazelle from the lion it is much more efficient for signs of a lion to trigger the flight response directly than to program the gazelle with some sort of realization that lion signs hint at a future situation that would be unpleasant and to try to figure out that running is the best way to avoid unpleasantness.
Evolutionary theory suggests that it is reasonable to expect complex creatures to acquire a certain amount of altruism. A parent tht cares for its children - not as a means for an end but as ends in themselves - have children that grow up to have and care for their own children. Thus, we are more likely to be the descendants of creatures with some amount of evolved altruism than from totally selfish ancestors.
Finally, egoism fails to account for conflicts among our desires. If all desires melted down to concern with our own pleasure or happiness, and there is a conflict, then we simply need to choose the option that gives us the most pleasure. If somebody were to offer an agent either one million dollars or one billion dollars, the rational agent takes the one billion dollars with no regret at all over having lost out on the chance to have merely one million dollars instead. Similarly, a person given a choice between ten units of pleasure or five will take ten.
However, our desires do not work that way. When we must make a choice between two options, we choose the one the option that (given our beliefs) will fulfill the most and strongest of our desires. However, we often regret the options that are closed to us. A person who must decide between going to a movie with a friend or staying home with her sick mother regrets the option he must sacrifice. This suggests that the agent has more than one desire. Both desires place demands on him and those that cannot be satisfied leave the agent with regrets.
The ethical egoist can accept all of these problems with psychological egoism and still say that of all of the possible desires only self-regarding desires have merit. Self-regarding desires are the only legitimate reasons for action that exist while all non-selfish motives are illegitimate and ought not to be considered. The objection here is that the ethical egoist cannot identify anything in the universe that justifies this distinction between legitimate and illegitimate desires. A reason for action is a reason for action - regardless of what it takes as its object.
Desirism is not an egoistic theory. It appears to be one only to the person who fails to distinguish desires in the self and desires of the self. All of an agent's intentional actions are motivated by desires of the self - if they were not motivated by her desires then they are not her actions. However, those desires are not limited to desires in the self.