Anarchy is the lack of government - or more specifically, the lack of governmental enforcement of rules. Traditional political philosophers, such as Hobbes, generally believed that this would result in violent disorder, but this is not necessarily the case.
Government, if defined as an organization that systematically enforces a set of rules, are not necessary for order. Most things can be controlled by social systems other then a government--for example peer pressure, the family, and religion.
The problem with starting an anarchy is that without strong organization large cities (and countries) will quickly fall apart. However, we have various examples of well-functioning anarchies. The !kung, along with other hunting and gathering groups, are recent examples of working anarchic systems. These societies tend to live in small groups of 20 or so individuals. They also have not stood up well in the face of large-scale governments trying to move into their territories. In recent memory, such societies have largely only survived in the harshest and most demanding of environments.
One of the main ideals of modern anarchy requires that no-one hurt anyone else. This includes any form of violence except possibly forced banishment in extreme cases -- if a person is uncontrollably violent, they can not be allowed to live in a system in which no-one else will raise a hand against them. Exactly what is meant by forbidding anyone to 'hurt' others changes from one view of anarchy to another, but murder and slavery are right out.
Generally, governmental power is based on the ability to enforce rules by incarceration or confiscation. If law-breakers were to put up any type of resistance a government could do nothing under an anarchic system. Therefore, as far as anarchists are concerned, governments are near useless (as far as law enforcement goes). Governments would not be able to enforce tax laws, so they would have to function either as non-profit organizations, surviving off of donations, or would have to functions as corporations, charging for the services they provide as utility companies do.
Military action against outsiders is usually considered okay under anarchist systems -- As long as enlistment in the army is voluntary. Only the hurting of members of your own society is un-anarchic. Weird.