Anarchism has a venerable history, going back to utopians and great thinkers of the 19th Century. Leo Tolstoy, Bertrand Russell, Emma Goldman were all sympathetic to anarchism in their professed philosophies, and the roots of anarchism go back to philosophers Henry David Thoreau and Peter Kropotkin.
The philosophy of anarchism grows from a belief in the goodness of human nature and the dangers of centralized authority. The idea is simply that local groups of people, left to their own devices, will govern themselves in their own way. Anarchists believe that centralized authority, which has been justified in the name of Rule of Law has historically been used for war, repression, and unreasonable restriction of liberties. Anarchists have historically been strong libertarians, and also communitarians.
Whenever the people rise up to ask for their fair share of the products of their labor, they have been tarred as "anarchists." The mainstream of Anarchist thought is militantly non-violent. But from the very earliest days, authoritarian governments sought to associate the word "anarchy" with chaos and with violence. This is a slander, but it is an old slander with a lot of sticking power. "Anarchist" is currently being used to describe looters and vandals.
This week, there are paid thugs infiltrating the urban demonstrations across America that are branding their vandalism with the circle-A symbol. It's classic behavior of false-flag violence.