In short, if you understand standard economics, you are a libertarian. Comparative advantage, immigration as bonus human capital, the inefficiency of government intervention in all spheres of life, particularly drugs and charity. Libertarians believe that what is economically best is best simpliciter.
An anarcho-capitalist is a radical economist, usually Austrian, who also holds the moral position that coercion is always bad. You can often see this phrased as 'the NAP,' the non-aggression principle.
(I will mainly be avoiding the cultural aspects. For example, most libertarians like gun ownership to the point they're suspicious of non-gun owners. (I don't own a gun, just a bow.) It seems that standard economics broadly supports freedom, and therefore libertarianism attracts the freedom-loving. Most libertarians think freedom is the highest virtue. None of these position are necessary or entailed. Occasionally the positions makes them ignore their own theory.
Anarcho-capitalists are frequently extremely cantankerous toward authority, and while this improves free thought, it causes disagreements for the sake of rejecting any particular ancap as an authority on ancap. I will be shamelessly promoting my own version. For example, I believe this cantankerousness allows ancaps to see the many flaws in standard economics, not only in being not-Austrian but also in its many bits and pieces of state-worship. Get paid by the state, be unduly sympathetic to the state.)
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The libertarian believes that the state - defined here as the monopoly on legitimate coercion - is either unavoidable or that a certain amount of strictly limited coercion is necessary for a healthy economy. Libertarians vary between hard minarchism to moderate minarchism. Between the night-watchman state and that plus rules against driving without a license and burning soft coal for heat. (See comments by the author.) Libertarians believe that the law is inherently coercive, inherently necessary, but that this coercion can be safely bottled up with objective rules, at least as long as the populace is moderately vigilant. For example, as long as there's a free press which seeks out abuses of power and reports them to their subscribers.
Libertarians, living in democratic times, believe that which laws the night-watchman state should enforce should be decided by popular vote.
Libertarians justify identifying the economy with human flourishing. Free humans work towards their own personal gain, which they are capable of achieving. However, other humans are roughly as competent, which means direct competition is costly; instead, humans naturally cooperate, resulting in exchange of value for value, making everyone richer. When value is exchanged, it is proof that humans are successfully working towards fulfilling their own values, and this can be measured, if crudely, by things like GDP.
For a different way of seeing this, a strong economy absent government favouritism makes everyone richer; richer people are more capable and powerful, and thus able to satisfy their values.
Free is defined as not interfered with by the state. Religion is a distant second, but in some times and places, religion can harness coercion much like the state and thus interfere with the exchange of value for value.
Standard economics proves that you cannot tax capital, or equivalently that progressive taxes are highly distorting, which means wasteful and counter-productive. The state should be funded by a flat income tax. In any case, taxes are way, way too high.
Some libertarian predictions:
Free trade is always to the benefit of both parties, or otherwise they would stop trading.
Immigration provides more labour, which means more wealth. It is best seen as free trade in labour - someone at home has to be willing to hire the guy to make it worth the move.
Arming and training average Joes would decrease crime, decrease the need for police, and even decrease death by guns.
Taking drugs may be a poor choice for some people, but it is their mistake to make. The state attempting to stop them is horrendously violent and destructive, and mostly futile in any case. Legalizing drug use has been documented at least once to reduce drug use. Despite price decreases. Revoking the USDA food pyramid and generally getting out of food-nanny Dodge would similarly increase public health.
Libertarians predict that the economy would rocket if economics was taken seriously. Here's (same as coal-link above) a moderately complete list of policies that should be rolled back. (Ctrl-f "ordinary Europeans" to skip the preamble.)
Lack of government intervention includes fiat money. Libertarians usually favour gold, but bitcoin works, indeed any currency that doesn't inherently depend on coercion.
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Anarcho-capitalists reject even the night-watchman state on several grounds.
While libertarians are sympathetic to public-choice theory, ancaps aggressively embrace and extend it. Public choice theory indicts all forms of government. It makes the tyranny of the majority seem like a mild pathology, barely a hiccup. There is no market failure so horrible that it won't inevitably be made worse by government intervention.
For example, the conversion of a democracy to a permanent bureaucracy is inevitable. Facing either term limits or an uncertain election in the future, the elected politician works toward legislation to create a sinecure for their buddies, who will then hire them after their term. This is hardly difficult, as their fellow elects feel the same way. Similarly, since state agents are selfish, they will try to arrogate all benefits to themselves. Since by definition of 'state' they have the power to do so, they will succeed. There is literally no point in having a government except to enrich government. Almost every case looks like the government creating the healthcare crisis it now wants to pretend to solve. Even if legislation manages to ban soft coal, coal was already going out of style. Legislation is a trailing indicator of good behaviour, and a leading indicator of only harmful behaviour.
I'll also mention the various historical records of social order achieved through voluntary, bottom-up processes, showing that the state is unnecessary. I can point to Iceland, Somalia (outside Mogadishu), a place in upland southeast Asia called Zomia, and I've forgotten one or two others.
However, strictly speaking, anarcho-capitalism is the ur-government, under which almost any social order can be constructed...presuming you can convince the subjects to sign on the dotted line.
There's currently one now, the Amish, who are 100% full-test anarchs by my theory.
"The ordnung is only accepted by, and binding on, members of the congregation when, as adults, they are baptized. [...] Prior to the ceremony, ministers offer the young adult the opportunity to back out, telling him that “it is better not to make a vow than to make a vow and later break it . . . .” [...] A bishop whose interpretation of his congregation’s ordnungen is at odds with what the members want is not subject to impeachment or a recall election, but he is at risk of finding himself with no membership."
It is taken as axiomatic that all coercion - defined here as one human imposing their values on another - is bad. As can be seen by the definition, for coercion to occur, one human must smother another's values, inherently reducing the total value of the world. I will note, but defer the proof, that this definition also justifies self-defence, such that self-defence against an attempted value imposition is not itself coercion. (Basically, defeating evil can't be evil.)
The anarchists' rejection of coercion has some surprising consequences. Did you know free trade, including free trade in labour, is illegitimate? It is the government deciding for you whether you want immigration or not. When two trade, either I'm one of them and it it's none of your business, or I'm not one of them and it's none of mine. Using the Amish model, it is none of my business what another congregation's Bishop says about immigration or trade.
Similarly, 'freeing' the slaves. The slaves were not given the choice to stay on the plantation. Perhaps none would have wanted to, but the abolitionists did not ask, they simply imposed their values on another thede.
Part of ancap's rejection of personal authority includes rejection of standard economics. When it says comparative advantage, ancaps say, so what? The market knows better than you. Better than any of us. If what you say is true, then the market will agree with you. If not, then the market will kick your ass. If the Bishop says immigration is bad, and it is, then the market will reward the congregation that bans it. Part of how you know coercion is bad is because it is unnecessary.
Naturally, ancaps are very suspicious whenever standard economics shows the state did anything good. (I've yet to find a case where the suspicion was not justified.) So-called 'public' goods are either mismanaged regardless or only seem public through sophistry and ignorance.
Due to the inherent moral position against coercion, anarcho-capitalists believe the police and army should be privatized. (Please note, not abolished.) There is disagreement about what, exactly, this would look like, and I'll be selfishly talking up my own position. Essentially, law needs to be opt-in, with the option of attempting to provide your own security. This allows individual variation on laws, which allows noise, which means experimentation and natural selection.
Anarchy is hands-off in defining human flourishing. If a truly free society does not want to grow economically, it does not have to. (The Amish grow in population but are decidedly slow in buying power.) Though, it is considered likely. Humans want to empower themselves, and when this comes through techniques and technology, it means economic growth. If nobody is being coerced, then by definition values are getting maximized.
An anarchist individual may hold any religion they want, but are subject to self-defence if it wants to impose itself on anyone else. Beliefs are property too.
Taxes, defined as coercive payments, are illegitimate. If you can't sell your service, then you don't deserve to be paid for it.
Anarchist predictions:
Violent revolutions are bad. Historical revolutions never stop their violence at self-defence. Second, while top-down order is bad, that doesn't mean disorder is good. {I like to say chaos = delta(power)} The market can only respond so fast to changes. When governments fall, they change too fast, and people die even if nobody is being beheaded on camera.
Voting, as inherently coercive, can never result in a free society.
As long as immigrants and the sponsors of immigrants are held accountable and responsible for the effects, the market will reach a healthy, pro-social level of immigration.
Being the ur-government, anarchy can accomodate fiat money. It's between you, your bank, and your trading partners; none of my business. However, fiat money probably can't survive without coercive backing, because its whole reason for existence is inflation. If businesses can refuse paper, they usually will, and we would likely return to a historical bank note situation, with added cryptographic assurances.
How you use drugs is none of my business. It is between you and your security provider.
Whether you should be allowed a gun is between you and your security provider. Will they have to pay more damages to you if you have no gun, or to others you use the gun on incorrectly? They would know a lot more about that than I ever will. Similarly, if a bar requires you to disarm before entering, they take responsibility for your safety.
The market is made of people. Any solution government can think up, the market can also think up, except it isn't allowed to force you to pay for it.
One way for private law to work is by trying crimes to acts on individuals. If you leave someone alone, they cannot charge you with anything. However, if you interact with them, you must implicitly or explicitly agree not to interact in certain ways they would rather you didn't, on pain of dealing with their security provider. In return, they agree to not do certain things to you.
Many anarchists want criminal penalties to stop at shaming. I think it should be formal and contractual. Then, the penalties are definitively not coercive - you have agreed to them, and applying them is simply holding you to your word.
As for sudden violent assault, then you never agreed not to defend yourself. Equivalently, almost everyone values seeing the sudden assailant defeated.
In practice, we would have laws and courts almost exactly as we have now, perhaps even professional police, but the laws others would be held to when dealing with you would be decided by you, not by the court. The market would reward courts that defended their clients well, not those with 40 000 laws on the books. If for no other reason than it would be too expensive to try to trade with them.
It may turn out that the honour and reputation system for contracts is superior to violent tort enforcement. Someone will try it, and we'll find out.
