Here are a couple of Anarchist quotes I found lately, which I think help to explain what Anarchism is essentially and fundamentally all about.
If your object is to secure liberty, you must learn to do without authority and compulsion. If you intend to live in peace and harmony with your fellow-men, you and they should cultivate brotherhood and respect for each other. If you want to work together with them for your mutual benefit, you must practice cooperation. The social revolution means much more than the reorganization of conditions only: it means the establishment of new human values and social relationships, a changed attitude of man to man, as of one free and independent to his equal; it means a different spirit in individual and collective life, and that spirit cannot be born overnight. It is a spirit to be cultivated, to be nurtured and reared, as the most delicate flower it is, for indeed it is the flower of a new and beautiful existence. — Alexander Berkman (What Is Anarchism?)
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. — John Zerzan
And here's one I found some time ago by a famous Anarchist revolutionary, which I think basically and honestly explains the primary goal of Anarchism:
The anarchists are not promising anything to anyone. The anarchists only want people to be conscious of their own situation and seize freedom for themselves. — Maria Nikiforova
I hope to post many more I find on the net, be it in Facebook groups or Anarchist websites.