The idea is that anarcho-capitalists believe society is made safer more effectively by private police forces that compete with each other to serve citizens. A specific business model has never (that I know of) been specified, not because it's inconceivable, but rather because things could be handled a variety of ways. For example, you might live in a gated community that offered its own, internal police force, paid for by homeowners' association fees. In that case, your private police force would be tied to your location of residence.
(Note: There are a lot of good examples of private security like this out there in the real world today. It is more common in developing nations, but not at all uncommon in places such as universities in the United States, that form their own police forces paid for by tuition, grants, donations, and other funding streams.) Alternatively, you might live in an area served by multiple police forces among which you choose the one that offers the best protection/service at the lowest price. Here, you might subscribe to police services that can be invoked whenever you feel you need them, or perhaps you simply wait until you need to contact a police officer, and then choose a specific force's number to call. Or maybe you call a central dispatching agency that puts out a notice that someone is in trouble, and the police forces all compete to arrive first and gain your business.
As you can see, there are many possible ways that private police forces "could work," many of which presently exist in the world today, and are fully functional. So the first point I want to make is that it is not fair to merely dismiss the idea as wholly outlandish. Not only is it not outlandish, it presently exists and you have almost certainly availed yourself of its benefits at some point during your lifetime. (If not, give it a few years, it'll happen.)
The second point I want to make is that, since it is true that private, competing police forces already exist in the world today, as do public police forces, we cannot say that the two arrangements are mutually exclusive. The fact of the matter is that public and private police forces have learned to coexist peacefully with minimal occasional conflict. Therefore, the anarcho-capitalist position is not really "using private police forces," but more specifically, eliminating public police forces and letting the existing private police market expand in their absence.
Before I go on, let me summarize some of the rationale involved here. The problem anarcho-capitalists have with public police forces is many-fold. First of all, if the police fails you, you have no recourse. (More to the point, you only have legal recourse against the police "monopoly" if you survive whatever event we're talking about - which makes things worse.) Private, competing police forces theoretically address this problem by: (1) offering you more options, and therefore providing police with a profit motive to get the job done correctly; and (2) offering you a second police force to pit against the first.
Regarding your legal recourse, it is notoriously difficult to "win against the police" in court, because the criminal justice system is weighted in favor of itself. This brings us to the second objection to public police monopolies, which is the potential for (and reality of) the abuse of police power. We need not look very far to find examples of that. Anarcho-capitalist policing addresses this problem by (1) and (2) from the previous paragraph, as well as (3) the settling of legal disputes through mutually agreed-upon and mutually hired professional arbiters who do not hold bias toward one party or the other because they are paid to be objective.Don't you dislike monopolies? Why would a government monopoly avoid your scorn?
Competition creates better results than a government monopoly does.
We can see what the government does for us, but we cannot see what could have been had the government not had a monopoly on the use of force against us.
We can see the public schools, but becasue we can afford to pay for public and private schools we don't know what a society all private schools would look like. (Although evidence would suggest better reading, writing and math skills).
I say, bring it on, privatize everything.
You can count me amongst the anarcho-capitalists.
Ryan at Stationary Waves also points out a few problem, as he sees it with a wholly privatized police force.
I'd like to comment on one of them. (I'd have put this as a comment on your post, Ryan, had I been able to do so without joining Google+.)
The second objection is related to the first: How do we actually know that there is sufficient market demand for a sufficiently strong police force?"sufficient market demand" is a subjective term
What you or I view as "sufficient" depends on what our views are.
And the market is always right.
I may not agree with the market all the time (such as having tabloid magazines near the registers at stores) but the market tells us what we the people want. (To the extent that it is able with all of the government influence.)