Anarchism is the attempt to eradicate domination. This includes not only such obvious forms as the nation-state, with its routine use of violence and the force of law, and the corporation, with its institutionalized irresponsibility, but also such internalized forms as patriarchy, racism, homophobia.
Also it is the attempt to expose the ways our philosophy, religion, economics, and other ideological constructions perform their primary function, which is to rationalize or naturalize --make seem natural-- the domination that pervades our way of life: the destruction of the natural world or of indigenous peoples, for example, comes not from the result of decisions actively made and actions pursued, but instead, so we convince ourselves, as a manifestation of Darwinian selection, or God's will, or economic exigency.
Beyond that, Anarchism is the attempt to look even into those parts of our everyday lives we accept as givens, as part of the universe, to see how they, too, dominate us or facilitate our domination over others... Most fundamentally, I would see Anarchism as a synonym for anti-authoritarianism.
