Destructive and constructive anarchism, and how to smash the state
Strands of anarchist thought throughout their history can be divided into two strands - destructive and constructive. Destructive anarchism (not necessarily meant in the pejorative sense) seeks to destroy the state (and therefore authority) by attacking the institutions of the state. For example, Bakunin believed that storming institutions and burning records (for example tax and debt records) was enough to destroy the state - anarchy, the "natural" state of human kind, would then prevail.
Constructive anarchism seeks to carve out autonomous space outside state control within capitalist states. Many strands of anarchism fall into this category - from Proudhon's mutualism, where free credit provided by mutual banks and trading between small farmers and producers would cut capitalists out of the everyday lives of ordinary people, to the point where capitalism simply ceased to function, to the Zapatista-inspired movements which seek to occupy autonomous spaces and communes whilst being explicitly opposed to challenging for political power.
Although anarchists have traditionally criticised Marxists for being "statist", both these broad strands of anarchism fail to provide methods to destroy the bourgeios state. Modern bourgeois states cannot simply be attacked by small groups (as in destructive anarchism) - without successfully challenging the repressive armed wing of such states, such as the army and police force, such uprisings will simply be smashed. Even in Bakunin's day (?), this method was a failure - his coup in Lyon involved storming local government buildings, destroying records and declaring the end of the state, but was soon defeated when the state moved to defend itself, bringing in the army.
Aside from this obvious hole in destructive anarchism's logic, there are other problems with it - small insurrections are by their very nature elitist, and the idea that anarchism is the nature order, suppressed only by the existence of a state.
On the other hand, constructive anarchism also fails to challenge the state. Anarchists like Murray Bookchin believe in building holes into the state by opting out - "hollowing out" autonomous spaces where people live by their own rules, outside state authority, until the state ultimately collapses as it is undermined from within.
There are many problems with this, not least that it is often simply not revolutionary - carving out your free anarchist space within society becomes a lifestyle project rather than a serious challenge to capitalism. It's also almost impossible for many people to remove themselves from capitalism - it may be possible for small groups of politically conscious people to squat or buy their own spaces and run them by their own rules, perhaps even becoming self-sufficient with no need to be consumers, but this isn't a strategy for most people. The means of producing the goods humans need, from shelter to food and clothing are not controlled by ordinary people; they're in the hands of the capitalist class, and therefore it's practically impossible to survive without becoming consumer in some sense under capitalism. Equally, most people need to work for wages in order to survive - setting up free autonomous communes outside of capitalism implies not working, and very few people have the means to be able to afford this.
In reality, very few anarchists involved in "constructive" strands of thought take it to its logical conclusion and opt out of the state entirely - they work and buy food, and their political projects involve organising community cafes and squatted social centres as free discussion spaces for activists. Whilst these aren't bad things in themselves, they're hardly revolutionary - on the small scale and limited way they exist they pose no challenge to capitalism or state authority, and as explained above,larger scale, more "extreme" forms of opting-out are utopian - the kind of numbers needed to make an impact on the state, will never be able to participate.
So despite all the anarchist claims to anti-statism, neither of the broad strands of anarchism adequately challenge the state. However, in order to understand why we need to challenge the state in the first place, and what we'd replace it with...
