The definition of anarchism is one of the ticklish problems here, not least because there are so many varieties, some of which are mutually incompatible, while others often overlap. Anarchism is chiefly a leftist (or post-leftist*) system of thought, as with anarcho-communism or anarcho-syndicalism; but there are some descendants of the classical liberal and capitalist school of thought, while others eschew the whole business in favor of a totally primitive outlook.
Yeah, I'm this guy. I'm a little disappointed in me too.
The defining trait of anarchism would be rejection of the state. The socialistic anarchist who wishes to replace the state with self-governing, egalitarian economic communes, the anarcho-capitalist who wishes to replace the state and its operations entirely with private contracts, the anarcho-primitivist who wishes to replace the state with a return to prehistorical conditions of life -- all agree in considering the state an undesirable and unnecessary thing.
This doubtless sounds preposterous. And it must be admitted that many of the forefathers of anarchy were, to speak frankly, surprisingly bad reasoners. I've been reading some of the essays of Emma Goldman, one of the most important anarchist theorists of the early twentieth century, and was a little put out when, after page upon page of righteous railing against the injustice of the state and of the class system, she then spoke of the very concept of free will as a ridiculous idea. How anybody proposes to have human responsibility without free will, or any idea of justice apart from human responsibility, I don't claim to know. But I do not consider anarchism as such to be beholden to its prominent theorists (and I am confident that they would agree with me; they were as consistent as that in any case).
In her autobiographical book The Long Loneliness, Venerable Dorothy Day, herself a Catholic anarchist, quotes at length from Bob Ludlow, a fellow Catholic anarchist and a significant early contributor to the Catholic Worker, whose words are a more or less ideal expression of the views I've come to hold:
"Both among Catholics and anarchists ... a great deal of misunderstanding comes about by a confusion of the terms State, government and society. Father Luigi Sturzo's book Inner Laws of Society is the best Catholic treatment of the subject I have read. He brings out the point that the State is only one form of government. When you analyze what anarchists advocate (particularly the anarcho-syndicalists) it really boils down to the advocacy of decentralized self-governing bodies. It is a form of government.
"The confusion results because some anarchist writers use the term government as synonymous with the term State and make the categorical statement that they do not believe in government, meaning by that the State.
"The State is government by representation (when it is a democracy) but there is no reason why a Catholic must believe that people must be governed by representatives -- the Catholic is free to believe one way or the other as is evident from St. Thomas' treatment of law ... St. Thomas states: A law, properly speaking, regards first and foremost the order to the common good. Now to order anything to the common good belongs either to the whole people, or to someone who is the viceregent of the whole people. Hence the making of law belongs either to the whole people or a public personage who has care of the whole people ...
"Anarchists believe that the whole people composing a community should take care of what governing is to be done rather than have a distant and centralized State do it. You can see from the quotation from St. Thomas there is nothing heretical about such a belief. It certainly is possible for a Christian to be an anarchist. ...
"Our Lord taught us to pray 'Thy kingdom come on earth as it is in heaven' -- in other words the nearer earthly government approximates what things are in heaven the more Christian it is. I do believe -- whether it can be realized or not -- that the anarchist society approaches nearer to this ideal than do other forms of government. As the Christian lives in hope so may we set this as the idea, towards which we work even if it seems as impractical as Calvary." -- The Long Loneliness, pp. 268-269