Anarchism is a socialist political theory which rejects the heirarchical organization of society. Heirarchical authority, that is, the sort of authority wielded by States and by capitalists, is seen as contrary to the well-being of the individual and, moreover, unnecessary to the function of society. At the core of anarchist thinking is the fundamental belief that no one has a right to exploit or to oppress another person.
Anarchism is popularly touted as a violent or terrorist movement that seeks to plunge our happy consumerist world into utter chaos. This is pure nonsense.
Anarchism posits that the authority of the State, whose power is the monopoly of violence and therefore coercion, is unjustified and oppressive. Further, anarchism holds that capitalism -- that is, an economic system based on private property and wage labour -- leads to widespread poverty, exploitation and the marginalisation of people in favour of systemic interests.
Anarchists distinguish between the authority inherent in a person due to her talents or wisdom and the "authority" assumed by a person simply because he holds the superior role in an arbitrary hierarchy. Anarchists don't presume that society ought not to have structures, merely that the structures society does have should be based upon inherent, or natural, authority (Starhawk might call it power-from-within) and direct, participatory, democracy.
Anarchists believe that both the individual and society are best served when no-one dominates or rules over another. Anarchists tend to believe that social organization based on voluntary association, participatory and co-operative economy and a culture of individual empowerment and distrust of authority will lead to a more free and egalitarian society.