It's easier to say what they don't want. Unlike other political movements that define themselves by what they support—like republicanism, fascism, or monarchism—anarchism is defined by what it rejects. Namely, any authority that's imposed from above. Instead of a formal state with police and laws, anarchists envision a society in which small groups govern themselves by consensus. The functions that we now associate with the government, like mail, defense, and education, would be handled on a cooperative basis. The anarchist paradise would look very much like a set of independently run communes.
Small personal possessions would be OK in an anarchist state, but there would be no private land, and no one would privately own businesses or manufacturing equipment. Those are tools for accumulating wealth, which contradicts the movement's egalitarian values. Some anarchist philosophers reject the idea of money completely, imagining warehouses from which goods would be distributed on the basis of need. Others begrudgingly accept the usefulness of hard currency. Josiah Warren, the first American anarchist thinker, was slightly more imaginative: He proposed a system to pay people with certificates indicating how many hours of work they did. They could exchange the notes at local time stores for goods that took the same amount of time to produce.
Fomenting revolution through violence is a touchy subject among anarchists. Most of the movement's leading lights rejected the practice. (Emma Goldman, who conspired to assassinate industrialist Henry Clay Frick in 1892, was an exception, but she later disavowed those tactics.) Mainstream adherents have distanced themselves from the terrorists working in Italy, referring to them as nihilists rather than true anarchists.
But the intellectual movement has always been plagued by a militant wing. Some anarchists, taking their inspiration from the murder of Czar Alexander I in 1881, advocated "propaganda by deed"—the use of a spectacular act of violence to incite revolution. Anarchist publications celebrated the invention of dynamite, since it was so well-suited to this purpose. (In the anarchist Haymarket Square Riot of 1886, several policemen were killed when someone hurled a dynamite bomb into the crowd.) * German immigrant Johann Most urged American workers to arm themselves around the turn of the 20th century, and anarchists in New York and Illinois formed their own rifle clubs. It's not clear, however, whether the groups were planning an insurrection or simply defending themselves against regular attacks from the police.
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Explainer thanks Randall Amster of Prescott College, Jesse Cohn of Purdue University North Central, Tom Goyens of the University of Virginia, and Dana Ward of Pitzer College.
Correction, Dec. 30, 2010: This article originally misstated the date of the Haymarket Square riot as 1940 and said the bomb killed several policemen. The bomb killed one policeman, and six more were killed by gunfire in the aftermath. (Return to the corrected sentence.)