About

an·ar·chism/ˈanərˌkizəm/

Noun:

Belief in the abolition of all government and the organization of society on a voluntary, cooperative basis without recourse to force or... Anarchists as a political force or movement.

What is Anarchism?

Anarchism is "The theory or doctrine that all forms of government are oppressive and undesirable and should be abolished." [The American Heritage Dictionary of the English Language, Third Edition] Etymologically, 'anarchism' comes from Greek words translating as "an" (without) + "arch" (rule or ruler).The definition above is the most general of various definitions put forward by philosophers and political theorists. Naturally there are variations in formulation among the many diverse branches of anarchist thought. But a perusal of the literature makes it clear that all branches, from the ancient Chinese Taoists through the classical socialist anarchists to the modern anarcho-capitalists, agree to this core point that anarchism is anti-state.
For any political philosophy, there are two fundamental questions to answer:

1. What is the proper purpose of the State.
2. What is the proper extent of the State.

Anarchism answers the first question with 'There is no proper purpose.' and the second question with 'none.' While anarchism is the radical absolutist position on the extent of the state, we can learn much from those who are close to the same ideology. In particular, there is a large body of literature by people supporting a strictly limited State. A government restricted to providing police, judicial and defense services is called a minimal state. Those supporting such a government areminarchists. The term libertarian is used to denote the general anti-statist end of the ideological spectrum, and should be understood to include both minarchists and anarchists.Beware the provincialism of political writers who would expropriate the word "anarchism" for their own particular vision of a stateless society. Examples of sectarian definitions: Between Anarchism and Libertarianism by Jeff Draughn, Libertarianism: Bogus Anarchy by Peter Sabatini, and the Infoshop Anarchist FAQ.

General Classification of Political Ideologies

While all varieties of philosophical anarchism oppose the State, there are variations on how precisely "the State" is defined. Furthermore, different branches subsume the State in a more general concept of "authority", and this concept also varies. Agreement on the nature of the State (unnecessary) does not preclude disagreements on e.g. the nature of property, the provision of legal or security services, etc. Arrangements that seem totally voluntary to one faction might be deemed blatant coercion by another. The major area of dispute is over the question of property. There are anti-propertarian collectivist (socialist) anarchists on the one hand, and individualist (capitalist) anarchists on the other. This question of 'What is the nature of property?' is, of course, important to all politico-economic theories. The basic ideological map looks like this:

Map of Political Ideologies

This two dimensional model is familiar to many libertarians, but may be new to some readers. The ancient one-dimensional left-right ('liberal-conservative') model is insufficient to address the modern political landscape. One might argue that it has always been lacking, since it addresses only one of the two fundamental questions about government. ( a. What is the proper purpose?, and b. What is the proper extent?) The earliest use of the terms 'left' and 'right' as political descriptions was by European parliaments in the 18th century. Typically the aristocratic and monarchist representatives sat on the right side of the chamber, while the anarchists and reformers sat on the left. Clearly left-right usage at that time answered the 'extent' question, and would correspond to a vertical libertarian-authoritarian projection of our modern 2D model. Over time, left and right have shifted meanings. In the US today, left-right deals with purpose rather than extent. Thus it could be thought of as a horizontal projection of the 2D model.
To see similar 2D political models, take the World's Smallest Political Quiz, or check out the essay Between Anarchism and Libertarianism by Jeff Draughn.
The 2D model seems capable of handling a wide variety of political ideologies. In the following illustration, some positions have been marked.

Ideological Map with Selected Positions Plotted

The map above allows us to classify political ideologies, but it does not give any idea of the compatability between them. For example, the square map shows the same distance between communism and classical anarchism as there is between classical anarchism and anarcho-capitalism. Since anarchists agree on the proper extent of the state (none), and since the purpose of the State becomes less and less significant as the extent of the State decreases, the conflict regarding purpose withers away to the extent anarchism succeeds. In other words, the main reason that the treatment of property (in post-scarcity environments) is a conflict is that the power of decision is held by the State. Were such property matters decided and arranged voluntary, such conflicts would be trivial. In my opinion, there is room for diversity and experimentation in property arrangements. Syndicals and corporations, coops and single-propriators, and families extended, nuclear and communal could all coexist in a stateless society.