I've read a lot about anarchism but I failed to understand how the rules are enforced. Since anarchism refuses any form of vertical power, there is no police in anarchy, but there may be free associations. I suppose that every agreement is taken just by who accepts it, so the rules may be different for every free association. But in case there is some quarrel, how it is managed?
When having discussions on Anarchist/Anarcho-capitalist forums, I've come across this issue a couple times.
The most common theory, is that the people with a dispute will go to a third party(an arbitrator), who will resolve the dispute for them. If one party decides not to follow the Arbitrator's decision, than the community would consider that person to be a dishonest person, and just not do business with them. In short, they would ostracize that person.
In the case of a particularly troublesome person who is a danger to others, they would use force(violence), or they would hire a mercenary to use force on their behalf.
Arbitrators would be reliant upon reputation, so if an arbitrator was known to make biased or unfair decisions, than the idea is that people with disputes would decide not to use that particular arbitrator.
If you want to learn more, you can read the works of Murray Rothbard. I personally have barely read any of his works, but Anarcho-capitalists seem to worship him.
In an Anarchy, quarreling parties would be unrestricted to come to a lawful agreement. If examples where no strong central government are taken into account (Libya, central Africa, Afghanistan) what would most likely happen is that gangs (basically warlords) would form, and they would use force to get what they want. Individual citizens would ask the gang they live under, or would have to take care of the problem themselves however they chose.
This is exactly why Anarchy is a horrible system of government.
I often like to point to the workers councils of Hungary in October–December 1956 as a "pre-figurative" form of life without a state. Two examples of internal quarrels: many managers were instantly sacked by workers councils. Some of the replacement managers were subsequently sacked either by a council executive or by a workers council in general assembly. Additionally, council delegates to the Central Workers Council of Greater Budapest were recalled on a number of occasions for failing to meet the demands of their delegating council.
Additionally the CWCGB sought assistance from the Lawyer's council for the development of a constitution.
Generally, these social forms have sat like the British parliament has at times, as a simultaneous executive legislature and judiciary. The idea of "permanent executive session" for workers councils has been a commonplace in the history of these short lived institutions.
Sources: Any thing written by, or edited by, Bill Lomax, such as his sourcebook in English on the Hungarian councils.
Conflicts that are not peacefully resolved by their participants can be escalated to private defense agencies.
The theory is, because violence is expensive, PDAs will tend to negotiate and thus avoid violence in most cases.
Unlike with governments, individuals would be free to pick one or multiple PDAs without having to physically move. That includes being free to not pick any, and just protect themselves.
It is my opinion, that anarchy has been falsely related to politics when in fact it had an origin that is primarily concerned with equality and more importantly justice. Anarchists encourage preparation and education against authority; anticipating conflict.
So I could rephrase your question as 'How are quarrels managed according to true justice?'
In the western world's justice and political system, you can buy yourself out of prison and you can pay for legal (and martial) defence by an attorney (or an army). This is unacceptable for the anarchist. An educated anarchist society would have no room for this sort of injustice. It would ideally have disagreeing parties form councils for conflict resolution, instead of punishment. Moderating and minimizing hyperbole and polemic for the sake of swaying a jury and "winning a case", rather encouraging agreement between the disagreeing parties and recognition of the other's needs.
This is not a trivial endeavour and can't be done quickly. It requires an understanding of the cognitive nature and how human interests function. We all learn at different paste: We process data and thereby learn information by responding differently to different methods. We talk the other people, when we need their help understanding something. This results in the exchange of ideas.
As anarchists we are fully aware that rushing justice is an insult to attaining true justice. Equality is more concerned with broadening our conscience and helping others do the same than with punishment. As anarchists we confront abuse by any means necessary, and by accepting the consequences. Violent behaviour such as punishment solves nothing.