Sebastien Faure in Encyclopedie anarchiste defined anarchism as the negation of the principle of authority.
Before any sensible discussion on anarchy can take, we have to distinguish between anarchy and chaos.
Some people equate anarchy with chaos, but there is order in anarchy, or to be more precise, multiple orders. Its not like in an anarchic situation people will start acting randomly, they will still be as rational and behavioral as before. But there will be less conformity.
Anarchy = no ruler - so you are free to do anything you like.
Sounds very appealing, until you realise, people can do anything they like to you at the same time.
There are some doubts to whether an anarchistic system would be stable. It would most likely revert to one of the other forms of government over time.
For arguments sake, lets just say anarchy is stable, Anarchy is a contradiction. The basis of anarchy, is that you are sovereign, that you own yourself, so no-one has the right to rule over you. Also you have no right to rule over others. You have full right to self determination.
But in admitting that that you have a right, you are saying that there is some infrastructure or rules to the anarchistic system. There are rules to the anarchistic system, that no other system may be put in place. That the system is built on the justice/ideal of self sovereignty.
Self sovereignty or self ownership, implies you have the right to your life, the liberty of your life, and the property derived from your life and liberty of life.
If there are rights, then they must/will be enforced. A justice system must exist.
People cannot enforce their own justice, as an unbiased arbiter and due process must occur.
Here is a series of videos to show how Anarchists think it could work.
Robert Nozick wrote a good book on this. "Anarchy, the state and utopia".