There are two rather divergent lines of anarchist thought. The first is broadly known as “left-anarchism,” and encompasses anarcho-socialists, anarcho-syndicalists, and anarcho-communists. These anarchists believe that in an anarchist society, people either would or should abandon or greatly reduce the role of private property rights. The economic system would be organised around cooperatives, worker-owned firms, and/or communes. A key value in this line of anarchist thought is egalitarianism, the view that inequalities, especially of wealth and power, are undesirable, immoral, and socially contingent.
The second is broadly known as “anarcho-capitalism.” These anarchists believe that in an anarchist society, people either would or should not only retain private property, but expand it to encompass the entire social realm. No anarcho-capitalist has ever denied the right of people to voluntarily pool their private property and form a cooperative, worker-owned firm, or commune; but they also believe that several property, including such organisations as corporations, are not only perfectly legitimate but likely to be the predominant form of economic organisation under anarchism. Unlike the left-anarchists, anarcho-capitalists generally place little or no value on equality, believing that inequalities along all dimensions — including income and wealth — are not only perfectly legitimate so long as they “come about in the right way,” but are the natural consequence of human freedom.
A large segment of left-anarchists is extremely skeptical about the anarchist credentials of anarcho-capitalists, arguing that the anarchist movement has historically been clearly leftist. In my own view, it is necessary to re-write a great deal of history to maintain this claim. In Carl Landauer’s European Socialism: A History of Ideas and Movements (published in 1959 before any important modern anarcho-capitalist works had been written), this great socialist historian notes that:
To be sure, there is a difference between individualistic anarchism and collectivistic or communistic anarchism; Bakunin called himself a communist anarchist. But the communist anarchists also do not acknowledge any right of society to force the individual. They differ from the anarchistic individualists in their belief that men, if freed from coercion, will enter into voluntary associations of a communistic type, while the other wing believes that the free person will prefer a high degree of isolation. The communist anarchists repudiate the right of private property which is maintained through the power of the state. The individualist anarchists are inclined to maintain private property as a necessary condition of individual independence, without fully answering the question of how property could be maintained without courts and police.
Actually, Tucker and Spooner both wrote about the free market’s ability to provide legal and protection services, so Landauer’s remark was not accurate even in 1959. But the interesting point is that before the emergence of modern anarcho-capitalism Landauer found it necessary to distinguish two strands of anarchism, only one of which he considered to be within the broad socialist tradition.