An Anarcho-capitalist is one who supports the theoretical form of government in which all things are privatized, including government functionality. It is the type of libertarianism that
Ayn Rand herself found ridiculous, so despite their similarities, they and the
Objectivists are mortal enemies.
In an Anarcho-capitalist system, rather than pay taxes to a government for police protection, taxes which are forced from you, you may voluntarily purchase such protection from competing protection agencies. Services we normally associate with governments, such as highways, would also be privatized, and people who own the land could build roads and allow or deny anyone they please use of them, based upon whatever criteria they wished. Privately owned toll roads would likely be common in such a society. Proponents point to the efficient nature of markets, and the likelihood that the services offered by private companies with a profit motive would be far superior to a government without one. They also point to much lower taxes due to these efficiencies. The primary and overriding argument is fairness, though. No one has the right to take my life, liberty or property without my consent, so founding a society and government on free association and voluntary trade solves this problem.
Critics attack them from several fronts. Some say competition between protection agencies would lead to violence when a crime is alleged by someone utilizing one agency with a victim utilizing another. The anarcho-capitalists counter that it would be in the best interest of the agencies to have extradition and arbitration agreements, and agencies which abused their customers or attempted to defraud other agencies would be ostracized by not getting future agreements, making them less attractive in the marketplace than competitors and eventually leading to bankruptcy.
Others say those who cannot afford to pay for protection would be exploited. I don't know of the arguments against that. Particularly, one argument I have never seen refuted is the assertion that children, who have no income, would be completely at the mercy of their guardians. If someone wanted to have babies and kill them one after the other, there would be no recourse. I don't know of the argument against that, either, but I am by no means an expert on the matter. Probably the most accomplished theorists among the anarcho-capitalists are the late Murray Rothbard, Walter Block, and Hans-Hermann Hoppe. All three can be read extensively at Lew Rockwell.com.