Anarchism is, to put it simply, the rejection of coercive authority and hierarchies on the political, economic, and social spheres.
Anarchism is, unlike liberalism or conservatism, not just an ideology, rather it is a family of ideas, methods, philosophies, and different
arrangements of society.
Anarchism is a method of reaching anarchy, this hierarchy-less society.
Anarchism is prefigurative, meaning its methods match its overall goal for society. In other words, its organizations and actions are hierarchy-less and decentralized.
Anarchism has many different branches, and many of them intersect each other into various forms of synthesis.
There are economically focused anarchisms such as anarcho-communism, anarcho-syndicalism, mutualism, or anarcho-collectivism.
There are socially focused anarchisms,
such as queer anarchism or black anarchism.
There are philosophically focused anarchisms, such as individualist anarchism (egoist anarchism) or religious anarchism.
There are politically focused anarchisms, such as insurrectionary anarchism or nihilist anarchism.
Many of these
intersect, meaning you can, and usually do, identify with one or more of these labels. Many anarchists historically used the term 'Anarchist without adjectives' due to the complications that come from inter-label squabbling.
To learn more, check out
http://anarchy.works.
But aren't anarcho-capitalists a thing, Teatime?
No. Capitalists are not anarchists because they condone coercive authority and hierarchy in the economic sphere. They are anarchists just as much as someone who wants a government or strict social hierarchies is an anarchist.