‘Anarchy’ does not mean ‘chaos’. Well, to be fair, it does, but anarchists have spent the last 150-or-so years trying to redefine it. Success in winning the wider population to our usage has been…mixed.
Anyway, as defined by anarchists, ‘anarchy’ means organised society not based on coercion. By ‘not based on coercion’ I mean that things like allocation of resources, organisation of labour, maintenance of infrastructure, and so forth rely on free consent, not (as in our own society) on the threat and use of arrest and imprisonment: individuals are free to order their lives as they see fit, as long as they do not violently endanger others.
This means that people collectively will have no more (but no fewer) rights than people individually – the right to defend themselves and organise and administrate themselves, to meet force with force and to withdraw services, but not to coerce anyone, not to use force for economic or social purposes.
Anarchists are neither lovers of violence nor pacifists. They simply believe that if something would be wrong when done by an ordinary human being, it is wrong when done by the state and its representatives.
Anarchism is also more broadly concerned with contesting hierarchies and power imbalances in general – those arising from wealth, from sex, from race, from any source.
Power imbalances cannot be entirely eliminated – if I know something you don’t, I have more power than you. But society can be designed so that, like the movements of molecules in still air, they remain so tiny and evenly distributed as to cancel out.
A hierarchical society is one arranged so that, like the movements of air molecules in a gust of wind, power imbalance combine with and reinforce each other, producing ever greater differences between people. The result is the creation of tyrants on the one hand and slaves on the other, and pressure on those in between to be simply a mix of the two.
Some posts of mine that discuss anarchism and anarchy: