First things first: anarchy is not chaos. In fact, the philosophy, which has many quite different factions, is based on the moral and practical goal of greater social order through voluntary interaction rather than through force. Anarchy is the natural social order achieved through the consensual interactions of individuals.
Anarchy is derived from the Greek word "anarkhos" , meaning "no ruler". Those who advocate anarchy are called anarchists. There are many different types, based on their idea of what type of socio-economic system they prefer to engage in with others, including anarcho-syndicalists, anarcho-capitalists, anarcho-communists, anarcho-primitivists, etc. A quick read on Wikipedia can sum up the differences. What's important to know about anarchists is that they all reject the notion that people can be owned by and, consequently, ruled by, other people. They do not believe that anyone has a right to the person or property of others, precluding the idea that some (governments) may use force against others (citizens) in order to bend those others to their will.
The anarchist's greatest enemy, therefore, is government. Governments are those groups of people claiming for themselves the right to rule over others by force. The anarchist believes that people are born with equal rights and such rule is imposition against those rights. Anarchists advocate the abolition of government in favor of the spontaneous order of natural human society and consider it naive to believe that if people cannot be trusted with their own power as individuals, that they somehow can be trusted with the centralized power of government.
Typically, people are woefully miseducated on the ideas of human nature, society and economy, making them fear the idea of a voluntary society, even though the alternative, government, has proven to be a whimsical, utopian idea based on a theory, socialism, that has been discredited by economics and has proven to be a disaster in practice. For this reason, it is assumed that voluntary society cannot exist. Herein lies the biggest stumbling block to the philosophy: reversing the years of misinformation and replacing them with more factual, logical understanding.
The assumption for most is that government must exist to stop bad people from doing bad things. Anarchism disagrees. It should be clear that, not only does government fail in this, but it acts a vehicle for bad people to do really bad things like war, genocide, slavery, etc. Anarchists understand that security is a service that is subject to supply and demand, like any other. Therefore, it cannot be supplied effectively, efficiently or in a manner consistent with the desires of the customer by a coercive monopolist like government. All people desire to be secure in their person and property to a far greater degree than they desire the person or property of others as a general rule. Entrepreneurs realize this and will always offer such security in a competitive environment, consistent with the desires of its customers. If the customer is a syndicalist, they are free to procure security from a syndicate. If they are communists, they can choose to procure security from the commune. If they are capitalists, they are free to procure security from a company. All these entities are truly held accountable for abuse by their competitors or ideological alternatives. Governments are not.
The way to tell a true anarchist from a fake is that the true anarchist is never willing to initiate force against the person or property of others. He will never support government, nor will he require others to participate in his preferred ideology. The anarchist is not a lazy busybody willing to employ government to force his ideas on others. He relies, like every decent person should, on persuasion and, upon the failure to persuade, tolerance. All anarchists are libertarians, all anarchists are market anarchists. Ideology follows after those categories. All are libertarians because they must accept the non-aggression principle. All are market anarchists because they cannot seek their preferred socio-economic situation without market freedom.
Of course, there is much more to anarchism than the simple exposition above. Here are some links that get into the nuts and bolts:
basics of libertarianism 1
basics of libertarianism 2
contradictions of government 1
contradictions of government 2
contradictions of government 3
the anarchist solution 1
the anarchist solution 2
the anarchist solution 3
the anarchist solution 4
the anarchist solution 5
what are rights?
the failure of democracy
the constitution experiment
"Either the State for ever, crushing individual and local life, taking over in all fields of human activity, bringing with it its wars and its domestic struggles for power, its palace revolutions which only replace one tyrant by another, and inevitably at the end of this development there is ... death! Or the destruction of States, and new life starting again in thousands of centers on the principle of the lively initiative of the individual and groups and that of free agreement.
The choice lies with you!" - Peter Kropotkin