The “anarcho” prefix in “anarcho-capitalism” refers to anarchism. There is much debate over what the defining elements of anarchism are, but virtually all conceptions entail a society without a State (not necessarily without a government more broadly conceived). The State is that institution in a given geographical area which asserts a monopoly over the creation, interpretation, and enforcement of law. The laws it enacts supersede the rules made by any other person or organization within its jurisdiction. The State is the ultimate arbiter in all cases of interpersonal conflict, including those involving its own agents (the conflict of interest should be apparent). Lastly, the State reserves the unique legal privilege to lay taxes, i.e. to coercively demand payment for its so-called “services” and unilaterally set and alter the levels thereof. The aforementioned characteristics are inherent to all States no matter their particular type or configuration. Hans-Hermann Hoppe has this to say about the State’s monopolistic nature:
…among economists and philosophers two near-universally accepted propositions exist:
1. Every ‘monopoly’ is ‘bad’ from the viewpoint of consumers. Monopoly is here understood in its classic meaning as an exclusive privilege granted to a single producer of a commodity or service, or as the absence of ‘free entry’ into a particular line of production. Only one agency, A, may produce a given good or service, X. Such a monopoly is ‘bad’ for consumers, because, shielded from potential new entrants into a given area of production, the price of the product will be higher and its quality lower than otherwise, under free competition.
2. The production of law and order, i.e., of security, is the primary function of the state (as just defined). Security is here understood in the wide sense adopted in the American Declaration of Independence: as the protection of life, property, and the pursuit of happiness from domestic violence (crime) as well as external (foreign) aggression (war). Both propositions are apparently incompatible with each other. This has rarely caused concern among philosophers and economists, however, and in so far as it has, the typical reaction has been one of taking exception to the first proposition rather than the second. Yet there exist fundamental theoretical reasons (and mountains of empirical evidence) that it is indeed the second proposition that is in error. 
The “capitalism” suffix indicates the anarchic society specified is that of a free market capitalist variety. Under anarcho-capitalism, all scarce goods (to include land and other means of production) are subject to private ownership. Since capitalism simply refers to that economic environment where the means of production are privately owned, this system would be capitalist by default. It is, however, important to note that private ownership of a good does not necessarily entail it is owned only by a single individual. Two or more people may be partial/joint private owners of a scarce good (as is the case in corporations, co-ops, partnerships…etc.). To privately own something simply means having an exclusive right to utilize, occupy, or employ it, whereas there exist non-owners who have no such right. A “right” simply refers to that which one may justifiably employ force to defend or seek retribution for its violation.
The Problem of Social Order: Scarcity
The problem of social order arises when two or more people want to use the same good in incompatible ways. Goods which have the capacity for mutually exclusive usage, and whose demand exceeds their availability, are said to be “scarce.” It is due to such scarcity and the desire to avoid violent interpersonal conflict that property norms are developed. Hoppe expounds:
To develop the concept of property, it is necessary for [economic] goods to be scarce, so that conflicts over the use of these goods can possibly arise. It is the function of property rights to avoid such possible clashes over the use of scarce resources by assigning rights of exclusive ownership. Property is thus a normative concept: a concept designed to make a conflict-free interaction possible by stipulating mutually binding rules of conduct (norms) regarding scarce resources. 
In addition to helping avoid conflict, such property norms serve as the legal basis for adjudicating interpersonal disputes. That is, for determining who the victim(s) and perpetrator(s) are in any given (violent) conflict and what measures should be taken to most approximately restore the victim and penalize the aggressor.
Thus, the scope of political philosophy is confined to one simple question: when is the use of force justified? As noted earlier, force is only justified in response to rights violations, and one’s rights are determined by prevailing property norms. In essence, to determine when the use of force is justified and who the victim and aggressor are in any given conflict, one must discover who owns what.
The Private-Property Ethic
The particular property norm which serves as the core of anarcho-capitalism is the “private property ethic.” As alluded to earlier, this ethic states that all scarce goods (including land and other means of production) are subject to private ownership, given they are acquired via original appropriation or voluntary exchange. Original appropriation/homesteading states that the first user and claimant of a previously unowned good is that good’s rightful owner. Physical possession, transformation, emborderment, etc. are examples of homesteading acts. (It should be noted that creation has nothing to do with establishing ownership, as nothing is truly created nor destroyed but merely transformed.) Since this method of property acquisition awards ownership to the first user, it is necessarily conflict-free. Voluntary exchange is the second just means of property ownership. This entails that one came to own something via its prior owner voluntarily (contractually) transferring title to him. Being voluntary, this method is likewise free of conflict. Of course, alternative property norms exist, yet none fulfill the purpose of action norms as completely as the private property ethic. Hoppe explains:
Contrary to the frequently heard claim that the institution of private property is only a convention, it must be categorically stated: a convention serves a purpose, and it is something to which an alternative exists. The Latin alphabet, for instance, serves the purpose of written communication and there exists an alternative to it, the Cyrillic alphabet. That is why it is referred to as a convention. What, however, is the purpose of action norms? If no interpersonal conflict existed — that is: if, due to a prestabilized harmony of all interests, no situation ever arose in which two or more people want to use one and the same good in incompatible ways — then no norms would be needed. It is the purpose of norms to help avoid otherwise unavoidable conflict. A norm that generates conflict rather than helping to avoid it is contrary to the very purpose of norms. It is a dysfunctional norm or a perversion… With regard to the purpose of conflict avoidance, however, the institution of private property is definitely not just a convention, because no alternative to it exists. Only private (exclusive) property makes it possible that all otherwise unavoidable conflicts can be avoided. And only the principle of property acquisition through acts of original appropriation, performed by specific individuals at a specific time and location, makes it possible to avoid conflict from the beginning of mankind onward, because only the first appropriation of some previously unappropriated good can be conflict-free — simply, because — per definitionem — no one else had any previous dealings with the good. 
The Non-Aggression Principle and Self-Ownership
From the private-property ethic, one may deduce both the non-aggression principle (NAP) and the principle of self-ownership. The non-aggression principle condemns all acts of aggression as unjustified and criminal (i.e. violating property rights). Aggression, in this context, is defined as the uninvited initiation of physical interference with the persons or property of others or threats thereof. The principle of self-ownership stipulates that one is and can only be the sole owner of his own physical body. This reveals an important nuance in the anarcho-capitalist conception of private property.
Property rights over a scarce good are accorded to one who can demonstrate a superior objective link to it. Special consideration must be made in regard to one’s own physical body, as such a link is demonstrated by his uniquely direct control over it. That is to say, one controls or manipulates his body by will alone. For this reason, he is exempt from having to establish a property right over it through acts of homesteading or voluntary exchange. Conversely, one may control external goods only indirectly. He cannot control or manipulate external goods by will alone, but instead must employ the medium of his own body or some other good. Hence, the need for him to establish a superior objective link between himself and external goods through acts of original appropriation or voluntary exchange if he is to be recognized as their legitimate owner. 
Logical Errors of the State
If the State is charged with protecting the property of its citizenry, then it must be categorically stated that any attempt to do so will only end in contradiction. As stated earlier, an inherent characteristic of any State is that it must lay taxes in order to fund its operations. Taxes themselves are nothing more than threats to initiate uninvited physical interference with the persons and/or property of others if they do not hand over X amount of money. Recall, threats to initiate uninvited physical interference with the persons or property of others is considered aggression. Simply stated: taxation is theft. Thus, the State cannot make any attempts to protect the property of its citizenry without first violating it on a mass scale. Therein lies the State’s most glaring contradiction. Hoppe has this to say:
Rather, the state unilaterally fixes the rules of the game and can change them, per legislation, during the game. Obviously, such behavior is inconceivable for freely financed security providers. Just imagine a security provider, whether police, insurer, or arbitrator, whose offer consisted in something like this: ‘I will not contractually guarantee you anything. I will not tell you what specific things I will regard as your to-be-protected property, nor will I tell you what I oblige myself to do if, according to your opinion, I do not fulfill my service to you — but in any case, I reserve the right to unilaterally determine the price that you must pay me for such undefined service.’ 
Another issue involved with taxation is that it manifests the economic calculation problem. Since the State’s revenue comes from theft, as opposed to voluntary patronage, it cannot determine the most economic ways to employ its “services”, ways which would otherwise be indicated by profits and losses. The State is invariably in a position where it must make arbitrary, therefore uneconomic, decisions regarding what products/services to offer, where to offer them, how to produce them, how much to produce, what materials to produce them with, etc. More disturbing still is the fact that, had such resources not been wasted on Statist objectives, they could have otherwise been allocated towards productive market ends.
Of course, some will attempt to justify the State’s existence by claiming society is bound by a “social contract.” That for the same reason a restaurant owner may expect payment from a customer after he enjoys a meal, so too can the State expect payment for the “services” it renders. However, this is a faulty analogy for two reasons:
First, the State must commence in mass theft before it can provide anything. It would be more analogous to say the restaurant owner stole your money upfront but is willing to offer a consolation prize of some food. In actuality, one makes an individual choice to go to a restaurant and is only serviced upon his specific request. This is in distinct contrast to the State which offers a slew of unrequested “services” (some of which one may not even enjoy) and then expects payment in full.
Second, the restaurant owner presumably acquired his restaurant via original appropriation or voluntary exchange. As such, he is the legitimate owner of the goods and services he provides and is accordingly in a proper position to expect payment from his willing customers. In contrast, the State did not acquire the property for which it asserts jurisdiction via original appropriation or voluntary exchange (or if it did purchase property, then it did so with funds stolen through taxes), thus it has absolutely no legitimate authority.
Anarcho-capitalism Is Tolerant
In an anarcho-capitalist society, anyone can live in any way they see fit so long as they refrain from committing aggression as previously defined. If they would like to voluntarily pool their property with others to form mutualist enclaves, conservative covenants, or socialist communes, then such is their prerogative. The greatest variety of lifestyles are permitted under an overarching anarcho-capitalist legal system.
The Utopian Allegation
Anarcho-capitalism is not a utopian philosophy. No claim that crime or hardship will be eliminated is being made. Rather, what is being asserted is that an anarcho-capitalist legal system is superior to all others for the ends of peace, cooperation, and prosperity. That in this truly free market environment, the self-interests of individuals are harmoniously aligned with the welfare of greater society.
 Hans-Hermann Hoppe, “Property, Contract, Aggression, Capitalism, Socialism” in A Theory of Socialism and Capitalism: Economics, Politics, and Ethics (Boston: Kluwer Academic, 1989), 18. ↩
 For a rational proof of the private property ethic, NAP, and self -ownership see Chapter 1 of A Spontaneous Order: The Capitalist Case For A Stateless Society. ↩