The word "anarchy" is from Greek, prefix a, meaning "not," "the want of," "the absence of," or "the lack of", plus archos, meaning "a ruler," "director", "chief," "person in charge," "commander." The Greek words anarchos, and anarchia meant "having no government -- being without a government" [Angeles, Peter A.; The Harper Collins Dictionary of Philosophy, Second Edition, pp. 11-12].
As can be seen, the strict, original meaning of anarchism was not simply "no government." "An-archy" means "without a ruler," or more generally, "without authority," and it is in this sense that anarchists have continually used the word. For this reason, rather than being purely anti-government or anti-state, anarchism is primarily a movement against hierarchy. Why? Because hierarchy is the organisational structure that embodies authority. Since the state is the "highest" form of hierarchy, anarchists are, by definition, anti-state; but this is not a sufficient definition of anarchism. This means that real anarchists are opposed to all forms of hierarchical organisation, not only the state.
Reference to "hierarchy" in this context is a fairly recent development -- the "classical" anarchists did not use the word. However, it's clear from their writings that theirs was a philosophy against hierarchy, against any inequality of power or privileges between individuals. Bakunin spoke of this when attacked "official" authority but defended "natural influence," and also when he said:
"Do you want to make it impossible for anyone to oppress his fellow-man? Then make sure that no one shall possess power" [Maximoff, G. P., ed., The Political Philosophy of Bakunin: Scientific Anarchism, p. 271].
As Jeff Draughn notes, "while it has always been a latent part of the 'revolutionary project,' only recently has this broader concept of anti-hierarchy arisen for more specific scrutiny. Nonetheless, the root of this is plainly visible in the Greek roots of the word 'anarchy'" [Jeff Draughn, Between Anarchism and Libertarianism: Defining a New Movement]