What is anarchism?

Authentic anarchism is about some good ideas, but (unfortunately) also is apolitical and power phobic. The first anarchist was the 19th century thinker Pierre-Joseph Proudhon, he had differing intellectual periods in his life, some more interesting than others. The most interesting period was at the end of his life, when he laid the foundations for communalism, while advocating a federation of communes that would replace the concept of the Nation-State. But he was only very much interested in a negation of the existing economic (capitalist) institutions, and so, he never managed to develop very interesting alternatives for States and capitalism.
Anarchism then was, and still is, quite individualistic. There has been a tendency within anarchism that has tried to get rid of that situation, to get anarchism more organized, moral and rational, it's called social anarchism. Unfortunately, social anarchists have failed : anarchism experiences one of its most badly organized periods in history. Social anarchists try to overthrow coercive and exploitative social relationships, and are often so impatient that they start behaving in an irrational, flippant way.
So the gap between social and individualist anarchism has become quite bridgeable. Social anarchists try to replace exploitation with egalitarian, self-managed, and cooperative society forms. They want to obtain more liberties, even freedom. But while trying to achieve freedom, they often get disappointed, because these times are reactionary, not times at all in which we can achieve (social) freedom. So anarchists often look for scapegoats, that can be Socialists or Greens or former anarchists. Anarchists do not easily understand why anarchism is doing so bad, and remains so much unpopular.
The primary concern of the classical libertarian socialists and anarchists was negation, an opposition to the States and capitalism. They were against politics, they did not understand that politics was something else than statecraft. And so, they were not much interested in alternative, communal politics. They liked alternative economics, but they were thinking of power and politics as things that were bad. They did not want political empowerment, so they remained quite powerless.

And this only changed when revolutionary syndicalism got more influential wthin the libertarian part of the Left, about a century ago. The organized anarchists were opposing power but achieved empowerment, because they were advocating workers' assemblies, factory occupations and social revolutions. And before the rise of fascism, revolutionary syndicalism was quite succesful, a dream so many people liked to believe in, a fairy tale. This kind of syndicalism was about involvement in trade unions, mutual aid, and decentralization of decision making. In Spain, revolutionary syndicalism became so influential that it led to a Spanish revolution in the thirties.
Later, the social anarchist critique widened into a more generalized condemnation of domination and hierarchy : libertarian socialists like Murray Bookchin not only opposed classism but often also criticized patriarchy and racism. Many people within the Left realized what racism could lead to, they had experienced the World War holocaust and the victories of fascism. Bookchin also opposed the devastation of non-human nature and was a forerunner in making ecology and libertarian municipalism vital themes for the neo-libertarian left.
But anarchists did not always appreciate all of this renewing of the anarchist tradition. Many anarchists wanted anarchism to remain old-style or individualistic. And so, Bookchin and some other influential left-libertarian thinkers or activists left the anarchist movement, or were never willing to participate in it. Anarchism then further degenerated.