The use of the term “capitalism” by Anarcho-Capitalists, Austrian economists, and Objectivists, is quite unfortunate, because it has muddled the discussion beyond redemption, like most discussions between individualist and collectivist Anarchists. This is the major problem with trying to reclaim words: it takes too long to do so, and in the interim time you manage to alienate everyone else who is still using the previous definition.
Capitalism, as 99.9999% of the world understands it, is this statist economic system that we currently have in most of the Western world, a system where there is a synergy between the ruling class and the rich and powerful corporate elements within a society, where they lend mutual aid to each other as much as they can, at the expense of the general population.
As Market Anarchists, we oppose capitalism for many reasons, the most important of these being that we oppose the State as a whole. Capitalism, like socialism and communism, can be best described as a form of relationship between the State and corporations. In capitalism, the State is the ally of corporations, in socialism the State is the antagonist of corporations, and in communism the State takes over all corporations and thus fully assumes corporate power (which leads to the counter-intuitive conclusion that communism is merely the fullest expression of capitalism).
Under Anarchy, there cannot be “socialist structures” or “capitalist structures.” In Anarchy, there is only the market. There can be no question of whether Anarchists should support markets or not, as Anarchy simply cannot exist without markets as economic basis, and indeed all “non-market” proposals made by collectivist Anarchists (like the barter system or gifting) are merely different forms of markets. People are completely free to choose the sort of arrangements they want to live under, as well as to limit that freedom if they so desire. The only real limiting factor is people’s open-mindedness and willingness to go beyond what is “traditional” or “accepted” (such as the current monopoloid “law” system).
Capitalism and the market have about as much to do with each other as Creationism has to do with actual science. Capitalism is based on concentration of power, hierarchies of authority and specialization of labour. Markets have only one of these three fundamentals, specialization of labour. But the other two, the ones that create all the exploitation in capitalist societies, are not part of markets. For one thing, Anarchists are against hierarchies of authority, and that’s what causes a lot of the concentration of power in the first place. The State’s rules and influence accounts for the rest.
Anarchy is egalitarian and anti-hierarchy. Capitalism is classist. Anarchy is individualistic. Capitalism is collectivist.