Many anarchists, particularly within the Individualist (illegalist, egoist, post-left, insurrectionist, etc) tendencies tend to have a nihilist bent. We are not fooled by delusions of a communist or anarchist revolution, and if one were to even happen we have no faith that it would be a revolution in which we all got along afterwards and built some kind of homogeneous utopian Anarchy-land. However, unlike many revolutionary anarchists, we don't see this as a reason to stop fighting for our desires. Somebody could prove to us today that there would never be a revolution, and we would still fight. Many of us don't even desire a revolution, even if it were somehow possible, as we recognize this would just be replacing one world order for another.
Aside from the (non) revolutionary aspect, we are also moral nihilists and don't fall prey to the moral and dogmatic crusades of the left (including left anarchists).
Basically, anarcho-nihilism is the tendency to inject a healthy dose of nihilism (typically positive nihilism, generally) into your anarchy. And in my opinion you can't actually have productive or effective or sustained anarchy without this.
Renzo Novatore was an anarchist who talked a lot about his nihilism. Nihilist aspects can also be found in Stirner's writings. There is of course Nietzsche, who was not an anarchist but whom many anarchists find inspiration in.
Little Black Cart publish a handful of books dealing with nihilism from an anarchist perspective. Check out Boom!, Attentat, and blessed is the flame, which is a particularly fascinating book which looks at concentration camp rebellion through an anarcho-nihilist lens. Also check out the recent writings of the CCF.
The definition of nihilism can vary greatly, and some nihilists often don't resemble other nihilists at all. Take this quote from Novatore's "I Am Also A Nihilist":
I am an individualist because I am an anarchist; and I am an anarchist because I am a nihilist. But I also understand nihilism in my own way...
I don’t care whether it is Nordic or Oriental, nor whether or not is has a historical, political, practical tradition, or a theoretical, philosophical, spiritual, intellectual one. I call myself a nihilist because I know that nihilism means negation.
Negation of every society, of every cult, of every rule and of every religion. But I don’t yearn for Nirvana, any more than I long for Schopenhauer’s desperate and powerless pessimism, which is a worse thing than the violent renunciation of life itself. Mine is an enthusiastic and dionysian pessimism, like a flame that sets my vital exuberance ablaze, that mocks at any theoretical, scientific or moral prison.
And if I call myself an individualist anarchist, an iconoclast and a nihilist, it is precisely because I believe that in these adjectives there is the highest and most complete expression of my willful and reckless individuality that, like an overflowing river, wants to expand, impetuously sweeping away dikes and hedges, until it crashes into a granite boulder, shattering and breaking up in its turn. I do not renounce life. I exalt and sing it.
He conflates nihilism with negation, which are entirely different things. Traditional nihilism rejects the concepts of truth and universality. Negation is a logical connective that merely states that something/a proposition is not ("¬p" in formal logic). Negations are not incompatible with truths, in fact, they're objectively necessary to them. Nihilism is a subjective rejection of truths. It makes little sense when read from a classical nihilist understanding, beyond utter idiosyncracy, to use nihilism and negation interchangeably like Novatore does.
Nihilism in the sense Nietzche intended it is about taking control of your own destiny because no other authority could rival that of man's. This is what he meant by "God is dead and we have killed him". It means that in the absence of God, we must take responsibility for the world, because there is no higher being that can control us or tell us what to do and how to do it.
I think that nihilism or nihilist-anarchism acts as a foil to a lot of conventional anarchism. A decent amount of anarchist theory is derived from first principles (Human life is valuable, freedom is good, etc.) From those principles the rest of the theory is derived (If human life or life is valuable we should seek to protect it as much as possible and/or at all costs, capitalism exploits and devalues human life/life in general therefore we should reject capitalism etc.)
Nihilism rejects that. It denies most if not all first principles on the basis that they are really just arbitrary lines that don't have any backing. You can't philosophically prove that any life has intrinsic value or meaning. We can ascribe it value but that's still just an artificial attribution.
And it's not just that you can't prove that it has intrinsic value, it's also that declaring a sacred "right to life" is dangerous in itself:
Nihilism is the default of the universe.
What ultimately creates and justifies meaning and value in a human life are the myths and stories we are convinced by. You can see this in religion or sport most obviously. Just look at any sport you didn't grow up with and you'll be looking at the spectacle from the outside in because you just aren't convinced by it.
Because of this deeply-embedded conditioning, most people default to the only thing they believe to be anchored to something real; tradition. The myths that have conditioned us out of story telling and mythos-building are not convincing to a nihilist, who, for better or worse, sees through them.
A lot of anarchist nihilists no longer believe in the political program of socialism at all. Through constant re-evaluation of our values and desires, we no longer strive for democracy, equality, rights or social revolution. We simply wish to live uninhibited. Free of rulers, artificial constraints, morality and order. Without the burden of programs and social contracts and rules to govern us and our interactions with others.