Anarchy as a social, economic, or political philosophy – sometimes referred to as "anarchism" – is not actually one philosophy but a collection of many diverse philosophies. But almost all anarchists agree that the common thread of anarchism, the central premise of these philosophies, is a society without rulers. The word "anarchy" literally comes from the Greek words "ἀν-" meaning "no, not" and "ἄρχων" or "ἀρχή" meaning "ruler, power, authority". A ruler is a person or group who imposes their will on another person or group unsolicitedly, depriving the latter of the free exercise of their own will. "Unsolicitedly" means that this imposition was not consented to and was not a defensive response to such an imposition by the ones being imposed on.
So, anarchy is a philosophy or a society characterized by the absence of rulers. That's it. Within those limitations many options and variations are possible. And, since chaos, confusion, violence, disorder, and lawlessness usually entail one party enforcing their will on another against their will, philosophical anarchism is practically the antithesis of the common connotation of the word "anarchy".
Just to clear this up further, I'm going to go over a few things that anarchism, as a philosophy, is not.
Anarchy is not disorder. It is a lack of the use of coercion to force an arbitrary order on those who don't want it. People still have the freedom to come together and create their own order. In our society today, people do this quite often. They form organizations, businesses, cooperatives, swap and shops, clubs, and community action groups which can range from minimal levels of organization to highly organized and formal operations. In many cases, they do this without a ruler coercing them into forming or joining these organizations. People can organize themselves just fine without coercion, and there are innumerable everyday examples which testify to this.
Anarchy is not lawlessness. The absence of laws or rules is called "anomy". Anarchy can be understood as the forbidding of rulers, making it a law against rulership, which would put it in opposition to anomy. You can have law within an anarchist framework, you simply cannot have a person or group making up laws arbitrarily and imposing them on people who do not agree to them – or who do not first impose themselves on others. Murder, theft, rape, fraud, and any number of other crimes which victimize others are impositions of this very sort, and therefore are naturally forbidden within anarchy. If these things were permitted you would no longer have anarchy. It is also perfectly legitimate within anarchy for a group of people to come together and voluntarily decide they want to live within a set of rules which they mutually agree to.
Anarchy is not the absence of government, although this depends on how you define government. If government is defined as synonymous with the state – an entity which claims or exercises ultimate authority to impose its will on others over a given region – then anarchy is the absence of government. But if government is understood as a mechanism to steer, guide, organize, or coordinate a society then it is by no means mutually exclusive with anarchy. Anarchy is only incompatible with government that uses force, threats, fraud, coercion, or other means to rule and force its agenda on people without their consent.
Anarchy is not warlordism. Somalia is not an example of philosophical anarchy. It is the polar opposite of anarchy. If you have outlaws, warlords, and thugs imposing their will on people with violence, then you have rulers. That is not anarchy. Of all modern examples, the society that perhaps most closely resembles anarchy are the Kurds in Iraq.
Anarchy is not a utopian fantasy. Very few anarchists have the illusion that an anarchist society will be free of problems, abuses, crimes, and failures. We don't see anarchy as a panacea to cure all of the world's social and economic ills. We don't deny that an anarchist society could fail, collapse, or devolve back into statism. What anarchy is is a framework to give human beings back their humanity. It restores back to people their fundamental autonomous nature and attempts to give them the freedom to use this autonomy to create new solutions, to find what works for them, and to simply be human. Because the state, many of us would assert, is immoral and dehumanizing because it takes away the autonomy, self-determination, and individuality which centrally defines our human nature. It is not foolproof or failproof, but the worst case scenario of failed anarchy is simply the norm of statism which we have now.
Anarchy is not homogeneous. It is not one single philosophy, but many different philosophies which can sometimes be diametrically opposed to one another on some issues. There are socialist anarchists, capitalist anarchists, communist anarchists, individualist anarchists, mutualist anarchists, primitivist anarchists, transhumanist anarchists, and many other sorts. Sometimes these groups get along, and sometimes they do not. Sort of like statist political parties, which actually have a great deal in common, but still have bitter feuds with each other over small differences in their philosophies. There are movements among anarchists such as panarchism and synthesism which aim to bring these factions into unity, if not agreement, to work towards their shared goal: that there should be no rulers. These movements have recently been gaining ground as the abuses of the state are becoming more apparent and people are demanding change.
Anarchists do not want to destroy society. Though there are some extremist exceptions, most anarchists do not want to tear down civilization brick by brick. We don't want violence in the streets. We don't want chaos, panic, disorder, and lawlessness. We want to live together with other people in peace, without having to worry about a violent state deciding it doesn't like what we are doing, extorting us of our resources, or invasively interfering in our lives. We want to live, as much as possible, without the threat of harm for doing something that someone else disapproves of. We want to be able to organize communities, build buildings, create enterprises, work in a trade, get married, choose what we consume, defend ourselves from harm, and make our own life choices without asking for permission – so long as we aren't harming or endangering anyone else in the process. I think that's a rather reasonable thing to want.
And finally, switching back to positive statements, anarchists are all around you. Our numbers are growing. No one is sure just how many of us there are, as no one has attempted to do an anarchist census. Be we are here. If you live in a town of any significant size, then you probably live or work close to some of us. We are influencing your culture, your politics, and your thinking. We create many of the memes, articles, and videos you share on social media. We have spearheaded movements and societal change, both past and present. And we are raising our voices loudly in light of the current abuses of the state, its wars of aggression, and its increasingly massive corruption – things which nearly everyone is currently upset about to one degree or another.
Whoever you are, you probably have a lot more common ground with us than you realize. We know that you aren't going to agree with us on everything, and that's okay. We don't want to force our will on you, either. What we hope for from you is to open-mindedly work together with us, and with anyone who is willing, to try and solve the very serious issues we are dealing with currently. Because the politicians are not going to solve it for us. We, the people, have a responsibility to make our society and our world better and to stand up against the corruption, the wars, and the infringements that affect us all. And that's really what anarchy is all about – people willingly working together to make things happen while respecting one another as people.