"Webster defines anarchy as "a state of society where there is no law," and anarchism as "The doctrine of the abolition of formal government."... If we did not have government, we would be forced to invent it for self-protection. The only question, and the big one, is the proper role of government."
Fortunately, anarchists are not restricted to dictionary definitions in the pursuit of anarchy, which means, in practice, no rulers, not no rules. Anarchism is not a doctrine, but the body of thought and literature of those working for self-government, freedom from state oppression, voluntary cooperation and freedom of association.
Anarchists work to support local self-government, local economies, local social support systems. Anarchists support farmers' markets, cooperative child care, neighborhood assemblies, mutual aid, neighborhood watch, public libraries, housing cooperatives, sweat equity, public transportation, freedom of assembly, freedom of speech, and freedom of the press.
Anarchists are opposed to government imperialism, militarized police forces, Homeland (In)Security, corporate personhood and the corporate oligarchy, centralized political parties, standing armies, the centralized "global" economy, tax breaks for the rich and powerful, the overweening power and monopoly of force of a ruling elite.
The proper role of government is to carry out the will of the people. An anarchist government would consist of a decentralized network of federated assemblies carrying out the will of self-organized assemblies at the local level. An anarchist government would replace political and economic favor with decision-making by the people, locally, regionally and nationally. No central government would be able to wield economic or military power without the expressed will and consent of the people.
It is not anarchy that foists destruction, death, mayhem and chaos on the world, it is centralized government. The invasion and occupation of Panama, Grenada, Iraq and Afghanistan were not organized and carried out by anarchists, they were perpetrated by the central government of the United States, without the consent of its citizens, in support of global economic interests.
Cries of "Anarchy!" in response to vandalism and public violence are the unthinking, knee-jerk reactions of a public and media dominated by corporate and government propaganda, whose interests are served by denigrating popular dissent.
Anarchy is rule by the people, democracy in its truest form.