Most people do not consider the problem of government: it is not a person, nor a family, but a detached bureaucratic thing by which a few must rule over others they have not met. It does not consider the finer points of individual cases, but creates rigid abstract rules which inevitably come into conflict. For this reason, most governments spend their time in internal conflict over interpretation of rules, and inevitably oppress their citizens by forcing them to obey detailed regulations which fit an "average" citizen, yet apply to no actually living human being.
An anarchist is someone who agrees that civilization should exist, but believes that government is a parasite not necessary for civilization. Government both oppresses the citizens and takes from them the responsibility of making society work. Instead of taking it into their own hands, citizens are trained to sit back and call some distant acronymous agency to help them out. By this method, we domesticate ourselves and make even the best among us weak and passive. Even worse, we hand power to the bureaucrats, who are by definition people who could not succeed in actual work, and therefore take paper-pushing jobs so they have power over others.
The average person considers anarchy a state without order, but when looked at practically, it can be seen as a different kind of order. Centralized authority requires we all obey a single authority, but anarchy requires decentralized governments in which we are each our own authorities, and responsible to each other to collaborate and maintain what is needed for civilization. In centralized government, you have to convince a bureaucrat or jury that what you are doing is correct, but in decentralized government, you must maintain cooperation with your fellow citizens by showing them constantly that you are doing what is right. Centralized government is like watching television: you sit back and relax and pay attention to the show, but you are not actually part of it, and until it gets so bad you change the channel, you put up with its mediocrities.
Anarchists like to talk about "the n+1 problem." This refers to the fact that, in any government, you have a division between citizens (sheep) and authorities (herders). However, the authorities are sheep as well, since they are also citizens and are not given by nature any greater wisdom or ability. Because centralized government requires passive citizens obeying its rules, and for that reason needs people to enforce those rules, it can be said that for every n citizens there must be a certain number of authority figures. Here is where it gets complicated: because these authority figures are also citizens, and thus we cannot assume that they either will obey all of the rules, government must watch itself, in order to be fair. Thus for every n citizens, you need a certain number of watchers, and another number of watcher watchers. This adds up to the equation that for every n citizens you need at least n+1 watchers to keep society working fairly. Obviously, this is mathematically impossible.
What is missing here is trust. Trust, however, cannot occur when you have a society divided between people assumed to be doing wrong (citizens) and those assumed to be doing right (watchers). Furthermore, trust is nearly impossible when you have a government in Washington, D.C. which is trying to administer laws to places as different as Seattle, Washington and Mobile, Alabama. The only way one achieves trust is a society in which there is no division between citizens and watchers, and for that reason the responsibility to work well with one another is thrust back onto the citizens. When this happens, those who can trust each other exclude those who are irresponsible, forcing them to create their own social group to ensure their survival.
In an anarchist society, there is a lack of a single rule for all people, and therefore people cannot criticize actions simply because "they are illegal." However, actions which are unpopular because they are destructive or selfish will cause the person who committed said actions to be exiled from the society of his or her peers, without protection of law for actions that are "legal" but not ethical. Local communities are the focus of anarchist society, because informal decentralized systems encourage people to form social units only of those that they know. Anarchist society does not rely on "enforcement," or punishment of bad acts, but it relies on trust, or contined reinforcement of human relationships based on the day-to-day good that people do.
The moral construct of "good" and "evil" by which society lives is a materialist notion: it is designed to protect life and property, and does not consider the intent behind or results from an action, for example the necessity of driving away destructive people or confiscating property used to pollute rivers, as in the case of a dirty factory. Morality says simply yes or no depending on whether someone dies, or had their "rights" violated, or their property was taken away. Only secondarily do moral societies pass judgment over life and property, but by the very nature of morality, they are unwilling to do so on a large enough scale to have an impact. The death penalty is futile because a murderer stands good odds of beating it, and fining large corporations for their pollution is pointless because they will almost never pay anything commensurate to the actual damage done. Morality protects life and property, including of those who by virtue of possessing both will do untold damage to the citizens, the environment, and the public goodwill.
Anarchist morality is simple: do what you will. Those who are of like minds will congregate and form their own allegiances without formalizing them and thus detaching themselves from the task of building trust, and those who act in conflict with others will eventually find their will driven away or terminated by those who have a different agenda. This means that local communities will form according to the shared values of the individual wills involved; some communities will be dedicated to crime and drug use, and other communities will be intolerant of such choices and will defend against them. It is worth noting that no central government has ever solved the problem of crime, which is almost completely eliminated in local communities where everyone knows each other and have established a communal trust and values system, against which any transgression is clear and unwanted, whether it is "legal" or "moral" - or not. You cannot make enough rules to identify every destructive act, and those acts will differ from community to community.
Many people fear anarchy because they reason that, without some clear central authority saying right/wrong, people will act selfishly and destructively. The truth is that some people will always do that, and while they are protected under centralized authority, they are not in anarchies and therefore cannot get away with their legal and moral but unethical and destructive acts. Anarchy is not a revolt against morality, but as with government, a decentralization of it.
One great misconception about anarchy is that it is entirely an individual process, since the individual defines the values and rules by which he or she lives. This is not the case, as anarchy is not an impulse against civilization, but toward a decentralized civilization because it is inherently superior in design. Anarchists collaborate in an informal basis because of the values they share, upon which they act.
Instead of having our lives be organized by distant abstractions and rigid rules, anarchists prefer to connect with real living experience: the trust bonds that form between individuals (who, unlike in centralized systems, actually know each other and interact on a daily basis) to create active communities of collaboration, instead of passive communities in which arbitrary laws are enforced upon us by a barely-trustworthy bureacratic entity known as "government." If we want a world without the "abuse" of power, we have to recognize that power is abuse.
Some might construe anarchy as "radical" or "extremist," but when one recognizes that civilization is a natural impulse and government only a temporary means of asserting "control" for the benefit of centralized organizations like government and big business, it is anarchy that becomes natural and bureaucratic government that reeks of artificiality and extremity. Not all people need to be constrained by the laws that limit the lowest among us; in the name of avoiding that pitfall, our society rules us all inequally, ineptly. Anarchy is freedom from that delusion and the future for all who value experience over rules.