Anarchy is the idea of no government. It generally comes about when existing power structures fail or because, some argue, it is the nature of Man. In some ways anarchy can be the best solution to entrenched power structures that have to be undercut for true change to occur -- much like a farmer must till a field of weeds.
The problems/limitations with anarchy come to the fore when common resources get in the equation; that is, with items that others need (because they are scarce, for example) or no one created (land, water, etc.). Under such conditions, power eventually becomes central, either because population grows, resources become scarce, or technology amplifies the will of single individuals into an imbalance. Read up on "the tragedy of the commons".
Anarchy has no solution to this except use of force or other ad hoc solutions. Anything other than ad hoc solutions would create a proto-government which anarchy rejects.
Here's an interesting proof of concept: if anarchy was the eternal way, then YOU (a federation of united, yet independent, cells) would not exist. Do you suppose your cells should reject your authoritarian rule? We're talking about the difference between single-cellular and multi-cellular life. Can you grok it?
The American Revolutionaries started with a proto-anarchic state when they rejected England's rule. The Enlightenment and the Founders which created the United States Constitution were wiser than many appreciated, because they anticipated that Power would eventually need to faced openly and as honestly as possible to prevent either barbarism or dictatorships. The solution the American Founders came up with was the "separation of powers" to prevent concentration of power and the notion of "one person, one vote" where all are considered equal.
The alternative to the Civil Rule is to create a space-faring scenario, where people build until they can continue independently and indefinitely into space, never having to face the issue of living together. But, ultimately, that's not a solution either and only postpones the issue of living together -- eventually you get to the limit of energy in the Universe.