So, I wouldn’t necessarily say there is one single definitive movement. There are lots of different kinds of anarchists just like there are lots of different kinds of other kinds of leftists. There are eco-anarchists, anarcho-syndicalists, anarcha-feminists, anarcho-communists, etc. Personally, I find it more useful to view the many sub-tendencies of anarchism as tactics and trains of thought. To quote the late David Graeber, anarchism is something you do, not an identity.
But to get more into it… Anarchists broadly share an interest in dismantling hierarchy, but how and what that looks like differs from tendency to tendency and even individual to individual. Similarly, anarchists generally want to dismantle the state—we don’t believe in a dictatorship of the proletariat—because we believe the state is inherently hierarchical and inevitably leads to oppression and inequality. However! There are plenty of anarchists who attempt to distinguish between government and state. A state has that hierarchical component, whereas a government can be set up via federated municipalities as discussed by Murray Bookchin.
As an aside, the squabbling between Marxist-Leninists and anarchists really bothers me. They both tell each other that they’re stuck in the past, attempting to recreate failed political experiments and that neither has read the others theory… and like, Bookchin, Ocalan, and others, to me, represent the synthesis of anarchism and Marxism.