Tippie,
Don't know if I can give you an exact definition, but there are two kinds of political anarchism--left & right. The lefties are typical of the left politically--they want to destroy our government and replace it with a Marxist one. These are the guys you may have seen at the WTO protests causing so much destruction and duking it out with the riot police, etc. They're really hooligans, is what they are. Misguided, idealistic youths who think that they can achieve social justice by doing what Stalin did. Not too bright.
More interesting are the "right" anarchists. These are the more thoughtful (and peaceful) ultra-libertarian types who mostly just want smaller government or to find solutions that don't require government at all--such as contracts and voluntary associations. There's a group called the Voluntaryists for one.
The problem with the world "anarchist" is that it has been stolen and redefined to mean the same thing as "agitator". Most people envision a chaotic, surreal world in which everyone is fighting just to stay alive. But there's no reason why anarchy can't be peaceful, since all it really means is "absence of official government."
Take it for what it's worth. I've met some very intersting people who call themselves anarchists and I even agree with the ideals of some of them. I think we rely way to heavily on bureaucratic solutions and the welfare state. I'd like to see a return to individual responsibility.