On one level it is a kind of faith: a belief that most forms of irresponsibility that seem to make power necessary are in fact the effects of power itself. In practice though it is a constant questioning, an effort to identify every compulsory or hierarchical relation in human life, and challenge them to justify themselves, and if they cannot — which usually turns out to be the case — an effort to limit their power and thus widen the scope of human liberty. Just as a Sufi might say that Sufism is the core of truth behind all religions, an anarchist might argue that anarchism is the urge for freedom behind all political ideologies.
Schools of Marxism always have founders. Just as Marxism sprang from the mind of Marx, so we have Leninists, Maoists, Althusserians… (Note how the list starts with heads of state and grades almost seamlessly into French professors — who, in turn, can spawn their own sects: Lacanians, Foucauldians….)
Schools of anarchism, in contrast, almost invariably emerge from some kind of organizational principle or form of practice: Anarcho-Syndicalists and Anarcho-Communists, Insurrectionists and Platformists, Cooperativists, Councilists, Individualists, and so on.
Anarchists are distinguished by what they do, and how they organize themselves to go about doing it. And indeed this has always been what anarchists have spent most of their time thinking and arguing about. They have never been much interested in the kinds of broad strategic or philosophical questions that preoccupy Marxists such as Are the peasants a potentially revolutionary class? (anarchists consider this something for peasants to decide) or what is the nature of the commodity form? Rather, they tend to argue about what is the truly democratic way to go about a meeting, at what point organization stops empowering people and starts squelching individual freedom. Is “leadership” necessarily a bad thing? Or, alternately, about the ethics of opposing power: What is direct action? Should one condemn someone who assassinates a head of state? When is it okay to throw a brick?