Egoism is the philosophy of Max Stirner as described in his most famous work, "The Unique and Its Property" and expanded upon later in "Stirner's Critics". A 19th century existentialist philosopher, Stirner was one of the earliest known exponents of anarchy inside industrial Europe. Egoism stands apart from later workerist offshoots of anarchism like anarcho-communism and anarcho-syndicalism by refraining from glorifying work, the factory and other exploitative social constructs.
Egoism emphasizes the individual and their unique will and rejects any abstractions ("phantasms") and their influence ("haunting") on the actions, thoughts, feelings, and desires of the individual ("The Unique"). As such, Egoism is opposed to humanism, liberalism, statism, morality, ideology, work ethic, social custom, religion, tradition and other fixed ideas that are projected onto us by external forces. Stirner posits that The Unique pursue it's own interests, whatever they may be, free of any reservations born from phantasms.
Like most currents of post-workerist anarchy, egoism rejects the idea of mass social revolution, seeing it as a time of violent and unpredictable turmoil which could very easily give rise to new hierarchies that serve new tyrants who rush in to fill the power vacuum.
Instead, egoists and other post-workerist anarchists favor more evolutionary methods of making anarchy: A focus on alternative experiences and social experiments, as well as education and the demonstration of radical modes of living which can easily create anarchy in the world today, in the current time and place, serving the current population.
A lot of anarchists don't believe it's in any way desirable for individuals to wait for a pie-in-the-sky social revolution before they can begin to experience anarchy. Post-workerist anarchists have no qualms about celebrating life by fully-embracing alternative experiences and lifestyles outside of what is offered within the current social system.
Workerist anarchists are quick to demean post-workerists such as egoists, anticivs and green nihilists as "lifestylists" for not adhering to whatever workerist program their off-shoot of stateless socialism decrees as necessary to achieving revolution. Like all socialists, workerist anarchists would rather focus their energies on recruiting workers to their cause and growing their unions in the hopes that they (or more realistically their distant descendants) can accumulate numbers big enough to bring about their much-coveted socialist revolution.
Post-workerist anarchists want no part of any program designed by others to limit them, control them or curtail their individual desires in order to compel them to pursue a collective ideological agenda passed down by long-dead European philosophers who lived in a different time and place and had different ideals, customs and objectives than anyone living in the world today.
Egoists reject the idea that the individual should have to sacrifice for the benefit of the "greater good" and instead they posit that cooperation, the formation of social bonds, altruism and mutual aid are inherently desirable because these things benefit the individual as much as they benefit the collective. For this reason, Stirner advocated for a "union of egoists": Multiple egoists voluntarily associating with one another to fulfill a purpose, goal, or even to simply enjoy eachother's company; free of any coercion or obligation. It's essentially the earliest form of the anarchist concept of freedom of association.
Despite common misconceptions, egoists have nothing against relying on or working with others to achieve a mutually-shared goal. Egoism posits that kindness and charity is born from empathy, not morality. People give and help each other because it feels good for most people to do so, in this sense, what we call "altruism" is simply a side-effect of egoism.
Egoism embraces any act that is done out of the individual's desire to commit the act. If the act is born from obligation, it is not an egoist action. Egoism supports the individual doing exactly what the individual pleases - taking no notice of God, state, morality or society.
To Stirner, "rights" were merely specters in the mind, coercing us to act in a certain way in order to benefit externalities like the state. He held that society does not exist but "the individuals are its reality".