Presenting Stirner's philosophy left me with a choice between brevity and balancing all possible interpretations. I have settled for the former, so be warned: This is one man's interpretation. Go to the source to make up your own mind.
Understanding Stirner requires not only an appreciation of content and particular statements, but to a very strong degree an understanding of the structure of the work. According to Lawrence Stepelevich, the structure of The Ego and Its Own is modeled upon Hegels Phänomenologie des Geistes (The Phenomenology of Spirit). The Hegelianism in Stirner is not accidental, but rather essential.
Central to the Hegelian school of philosophy is that which is called Dialectics: Resolve dualisms by finding a third which explains/gives both sides. Stirner is a dialectical thinker in this sense. His main triad is that of Materialist - Idealist - Egoist.
Stirner follows up on Feuerbachs insistence that we must tie philosophy to the concrete individual, and later champions this insistence against Feuerbachs "Man", the species-being. Therefore chapter 1 in The Ego and Its Own, "A Human Life", is a statement of the dialectical development as it occurs in the life of concrete persons; as a child one is at the Materialist stage and fears the rod, as a youth one has made "the first self-discovery, Mind" and gotten back of the rod through Idealism, and as an adult also Idealism is seen as a kind of rod, and practical, selfish interest has taken over. This should, however, not take literally but rather figuratively.
Stirner then follows up on Cieszkowski by taking chapter 2, "Men of the old time and the new", to be a description of the same development writ large in history. The chapter ends with a section on his friends Die Freien, criticizing them as not representing the dialectical dissolution of the Materialist/Idealist opposition at all, but rather being "the most modern of moderns", i.e. the last Idealists.
"Likes are to be treated in the same way" is central to the Idealist stage. This is the basis of the Young Hegelian critique. By the inner dynamic of the critique, "likes" and "the same way" become ever-broadening categories, and "critique" must eventually turn on itself, collapsing under its own weight.
Stirner writes: If the presuppositions that have hitherto been current are to melt away in a full dissolution, they must not be dissolved into a higher presupposition again - a thought, or thinking itself, criticism. For that dissolution is to be for my good; otherwise it would belong only in the series of innumerable dissolutions [..]"
So this is the point from which Stirner's own philosophy starts, the collapse of Idealism and the need for a new synthesis. This new synthesis cannot, however, be an Archimedean idea-point outside the world, what Stirner calls a fixed idea. Thus a bit of care is needed when stating what Stirner proposes as a synthesis.
Stirner proposes the synthesis to be found in the interest of the unique - the egoist. This synthesis, qua isolated statement, puts Stirner in the same category as Thomas Hobbes, Friedrich W. Nietzsche, Dora Marsden, James Walker, Ayn Rand and Robert Nozick.
However, the kinds of egoism proposed by these philosophers is markedly different from Stirner's - with a partial exception for Marsden, who was strongly inspired by Stirner. The difference lies in the view of what I myself am, and the way egoism is arrived at.
We will take a brief description of some of these: For Thomas Hobbes, all that matters are external comparison of wealth and possession. Stirner's egoism is about the relation of the "I" and the object. In Stirner's synthesis, "I" am Subject, standing in relation to the object by my own will.
For Friedrich Nietzsche, there were set goals for the egoist to pursue. One should "create beyond oneself", create the Superman. Stirner, in contrast, focuses on consumption, the transitory, finite ego's appropriation of the world as its ("appropriate" in the same sense a student must make the literature he reads "his" in order to understand it well).
James Walker gives a biological description which more-or-less defines selfishness as anything the biological individual devotes its energy to, a mechanics of egoism. Stirner, in contrast, describes egoism as a possible chosen path.
Ayn Rand tries to prove egoism from first principles, putting "reason" plus a number of word-definitions - life (qua Man) and justice - as premises. The reply to the question of who is the just recipient of a man's labour, Rand claims, is that man himself. Acting according to that justice - seeing all values as instrumental to the fundamental value of life (qua Man) - is what Rand defines as egoism. Stirner, in contrast, does not "justify" his egoism, and Rand's "qua Man" is nothing but the species-being Stirner rejected in Feuerbach.
So, what is Stirnerite egoism?
As a preface to The Ego and Its Own, Stirner wrote a short piece Ich hab' mein Sach' auf nichts gestellt (I have set my affair on nothing; usually translated "All things are nothing to me"). In this piece, he shows how the Sultan, God, the Good etc. are not serving anything beyond themselves, but rather have set themselves up as the highest good to serve. Stirner writes: "I for my part take a lesson from them, and propose, instead of further unselfishly serving those great egoists, rather to be the egoist myself."
So indeed, he does not base his case on an imperative which he implores us to follow, but rather - seduces us by example. This is of focal importance if Stirner is to be consistent and not fall for the axe of his contemporary Karl Schmidt's criticism that Stirner is "making a new chimera" with his egoism.
Stirner's egoism then becomes more a therapeutic recipe for those who will accept it. Egoism for Stirner is just the following of one's own interests as the unique person one is. To somebody's "What are my interests?", Stirner would say that his interests are as unique as he himself, and that it would be for himself to find out. A repeated insistence would meet with only the negative answer Stirner provides in The Ego and Its Own, that one's interests and fixed ideas stand in opposition; that there is no Archimedean point of moral reference outside the values chosen by - the unique.
So "what am I?" This, Stirner spends the latter half part of his book exploring. That is, what are my relations like when they are not the material or natural bonds like filial loyalty or idealist relations like being "one and the same" as Citizen, Ragamuffin or Human? The key concept to answer this, is Eigentum - property.
"Eigentum", that which is owned, is for Stirner an expression of a willed relation. As a willed relation, it can be discarded at any moment - by will. Opposed to the willed relation is the bond, the "ought" and the "shall". These are simply relations that are not mine to dispose of, but which are given me from without - without also in the sense of an "essence" I must confirm to and cannot dispose of.
A particular case of such a bond is when you are not to let go of an idea. In Hegelian terms: When that thought is seen as exempt from and sacred to "the power of the negative". Such an idea is called a fixed idea. It is, in Stirner's words "An idea that has subjected the man to itself" - an idea that you are not to criticize.
The notion of "Eigentum" applies to relations with other people as well, and it is in this sense we must understand Der Verein der Egoisten (The Union of Egoists) which has confused and eluded the grasp of many commentators.
Let us have a look at the ways in which I can meet another person, from a point of view pertaining to the matter at hand.
Now, Moses Hess criticized Stirner's conception of what Stirner call "Verein der Egoisten" ["The Union of Egoists"] along the lines that in such a meeting, there would have to be one who did dominate and one who submitted to domination. That is, Hess imagined that "The Union of Egoist" would be a relation of the kind (2) described above. Now, (2) might describe a Hobbesian egoist. But can it describe "la derniere mallon de la chaine Hegelienne" (as Stirner has been called)? No, that is a bit too crude. Stirner did himself reply to this criticism by pointing to examples: Two friends playing with their toys, two men going together to the wine shop. These are of course not an exhaustive list of unions, and our man Stirner does indeed speak of unions consisting of thousands of people, too, unions uniting to catch a thief or to get better pay for one's own labour. More philosophically, Moses Hess describes a one-sidedness, and thinks it is a necessary one. What is then more natural than to apply a little dialectical reasoning to figure out what Stirner really did mean. I propose it is
Only after development has come to the understanding of the union of egoists does Stirner come to the ultimately important relation - the relation of me to myself. In the section entitled "My self-enjoyment", Stirner sets up mere valuing of life against enjoyment of life. In the former view, I am an object to be preserved. In the latter I see myself as the subject of all my valuing relations.
In this sense, Stirner can rebuke the question "what am I?" and replace it with "who am I?", a question which has its answer in this bodily person who asks the question. This is the "nothingness" of which Stirner speaks of as I. "Not nothing in the sense of emptiness, but a creative nothing."
My relation to myself is thus a meeting of myself as willer, a union with myself and a consumption - appropriation - of myself as my own.