Of course, I might be wrong — as I might be wrong about each and every belief I have — and my understanding of anarchism is always subject to revision based on new evidence and argument.
I do not understand what anarchists mean by "hierarchical authority" (or the related concept of Libertarian and right-anarchism of "initiation of coercion"). The best explanations I've read of these concepts boil down to the presence or exercise of authority or coercion the anarchist does not herself like. There's nothing at all wrong about disliking and condemning some particular exercise of authority or coercion. What I do not understand are the broad conclusions and anarchists seem to draw from the presence of objectionable authority and coercion and the principles they endorse in response to those conclusions. Purported explanations seem to rely on metaphor: anarchists, for example, don't want anyone to be "above" anyone else: in this case "above" is a metaphor; anarchists do not of course object to one person being taller than another.
I do not understand how is anarchism is different from capitalist democracy on the right and democratic communism (as opposed to Soviet or Chinese oligarchical communism) on the left. In their ideal forms, I don't understand the difference between left- and right-anarchism (libertarian socialism and Libertarianism/anarcho-capitalism) other than beliefs about people would actually behave under ideal circumstances; circumstances that differ so radically from present-day circumstances that I'm unable to grasp any evidentiary or scientific basis for their beliefs.
The closest I've come to a detailed and coherent account of anarchism is Systemantics, John Gall's witty and insightful examinations of how systems work and, more importantly, how they fail. The author concludes that systems are, by and large, not merely useless but counterproductive; we should simply do without systems wherever possible. Many of his principles conform to my own experience of building small- and large-scale systems; the principles that
A complex system that works is invariably found to have evolved from a simple system that works. A complex system designed from scratch never works and cannot be patched up to make it work. You have to start over, beginning with a working simple system.seem without exception, at least for certain definitions of "simple" and "complex".
On the other hand, the author presents very little evidence for his conclusions, and the evidence he does deign to present is cherry-picked, presented without context or detail, and his interpretation blatantly biased. For example, the author notes that the modern produce distribution system fails to deliver fresh, tree-ripened apples to the ordinary grocery store consumer, and concludes that the produce distribution system is therefore a failure.
More importantly, while the author describes at length the ridiculous pathologies of systems, he offers little in the way of explaining why systems pervade human society. His only explanation is there are some people, "systems-people", who are mean and bad and fat and dumb and have magically risen to positions of authority.
I'm obviously interpreting, but I see pervading all forms of anarchism the sense that those wielding the objectionable sorts of authority have magically gained this authority, almost as if they have landed in spaceships and enslaved humanity with superior technology. (Alternatively anarchists seem to reify authority from an abstract relationship between human beings to a concrete entity that can by itself be fought or eliminated.)
I do think that many of the systems actually in place — notably capitalism — are pathological and profoundly and deeply broken. And in conditions of massive oppression, exploitation and outright failure, I cannot condemn simply standing up and saying, "I'm mad as hell, and I'm not going to take it anymore," without offering a coherent alternative. I also understand and appreciate the value of people who are willing to oppose The System simply because it is The System, regardless of its particular characteristics. While I do not agree with Gall that systems are generally bad, with good systems rare and entirely accidental, I do think that any system that cannot tolerate and even encourage with good humor a certain amount "sand in the gears", i.e. opposition and resistance, is not a system suitable for human beings.
In other words, I'm not sure it's even important for me to understand anarchism. If anarchism labels an affinity group of people who simply want to oppose The System without worrying overmuch about the specifics, then good for them. Although it's not my personal affinity, anarchists in this sense must exist and to a certain extent thrive in any good system, especially a system of governance.
On the other hand, if anarchism really were, as many of its proponents suggest, a coherent, rational and practical political philosophy, then I do want to know about it and be rationally convinced.