There are two main schools of anarchism. Left-anarchists think we should get rid of government and private property and share everything, basically communism without a government. I have no idea how that would work. Myself, I'm a free market anarchist, which means I want to get rid of government and believe strongly in private property rights and laissez-faire capitalism. Free market anarchism (or Anarcho-capitalism) is based on the idea that there is no economic good which the public sector (government/ coercive monopoly) can provide better than the private sector.
UPDATE: *****I would encourage everyone to at least skim this article:
before posting, it is the basis for my arguments and outlines anarcho-capitalist theory in alot of areas that people often bring up, like law, the courts, and the police.*****
More general info on the Austrian school of economics at:
UPDATE: Found this site searching teh interwebs, I haven't read through all of it but it looks solid.