Egoism as meant in the sense of an "egoist anarchist" needs to be quickly set apart from other usages of the word "egoism".
An encyclopedia might describe egoism as taking three broad forms, "psychological egoism", which says that people always act in self-interested ways, "ethical egoism", which says that people should act in self-interested ways, and "rational egoism", which says that people who are rational act in self interested ways. I am not talking about these unless I specify one of these.
Egoism is also often confused with egotism, where one thinks they are more important to others than they actually are, and egocentrism, where one is emotionally disconnected from others.
Egoism as we use it in discussing Max Stirner's egoism or in egoist anarchism is something different. It can't be described in a short way without first explaining what egoism is defining itself against.
In his book "The Ego and Its Own", Max Stirner separates conscious from unconscious egoists. A conscious egoist is one who knows that the values they are pursuing are the ones that they want for themselves, where an unconscious egoist is one who pursues their own values while perceiving them as the values of some "higher" thing, a thing they feel is more important than they are. These "higher" things are often called "spooks" or "fixed ideas". Examples could be god, government, family, culture, country, society, humankind, property, marriage, honor, morality, gender, and others.
Such spooks have no consciousness of their own. Any ability to act, or any values that they express, are simply the expressed values of the people who participate in it. The values of family are the values of individuals about family, the values of family have no reality outside of those individuals. Only individuals experience want, value, and goals, these spooks, whether they are abstract principles or formal organizations, do not. But whenever any person holds the values of such a principle or institution above their own values, that principle or institution becomes a spook.
It's important to recognize that the egoist doesn't believe such principles or institutions are only imaginary and don't exist, only that the values they express are the values of those who participate in these institutions, and that those individuals' values have no inherent supremacy over any other individuals' values. The egoist recognizes that there is nothing special about them that would merit putting them above one's own values.
In this way, the egoist does not reject the values of other individuals, the egoist is rejecting the values of non-individuals. The egoist rejects every higher value than their own as nonsense. The egoist feels no allegiance to any spirit, any state, and family, any culture, any place, etc unless the allegiance benefits them for their own reasons, and then only insofar as it does.
So is Stirner's egoism a psychological egoism? Although Stirner never speaks of non-egoists, suggesting that all people always act in self-interested ways, this isn't actually what the ideas are about, and ideas of altruism or what non-egoism looks like, are simply not discussed. Stirner makes clear in later work that his egoism is not against any particular value, not against altruistic action, not against anything but sacred motivations for action. Stirner assumes at most that people act toward what they value, regardless of whether it serves them in any other sense than that they value it.
Is Stirner's egoism an ethical egoism? Clearly not, he doesn't create a new system of morality based on self-interest to show you what you should do, as Ayn Rand tries to do. If you see what he is showing you, that you and those around you may be controlled by spooks, his work is done, even if you decide that you like your spooks and want to keep them.
Is Stirner's egoism a rational egoism then? When he's not talking about rationalists or rationalism, he's usually speaking of "rational things" in quotes that suggest whether they are actually rational is less important than that they are called rational. So do people who are rational act in self-interestd ways? Given Stirner's approach to rationality, it seems he doesn't care about the answer so much as he cares that people who call themselves rational have fixed ideas about the whole thing.
To describe Stirner's egoism as any one of these things is either the opposite of the truth or is mistaking a precept for the concept. Stirner's self interest is not defined against helping others, but against serving ideals that have no consciousness of their own.
Then what is egoist anarchism? It's simply what happens when the spook of the state loses its authority, and one's own interests are pursued against it.