The Non-Aggression Principle gives us a basis upon which to analyze human interaction and make judgments concerning interpersonal disputes.
Many disputes are straightforward: Person A clocked Person B over the head with a tire iron and took his wallet. Person A violated the NAP, so Person B may rightfully retrieve his wallet, using force if necessary, and demand compensation for pain and suffering.
Some disputes take a bit more investigation to determine who is the aggressor, who is the victim, and what course of action may be taken to indemnify the victim. One such dispute, or rather class of disputes, involves women who find themselves carrying another life within their own bodies. The woman, as a self-owner, has the right to defend her body, using force if necessary, from violation. The new life within her, the human child, also has a right to not be killed, as per the NAP. (I'll not entertain here the so-called debate as to whether an unborn child is a "person" deserving of rights. A moment of honest reflection will reveal that, as the child is biologically alive, and its DNA is that of a human, the child is indeed a live human being, and attempts to draw distinctions between "humans" and "persons" is not only obvious sophistry, but has always been used to justify violations of the NAP.)
If we are working under the framework of the NAP, we are assuming that all human beings have the right to not be aggressed against, that we have the right to live our lives unmolested so long as we do not infringe upon the equal rights of others. So in this case we have a situation in which one person, the mother, is assumed to have the right to do what she wishes with her body, so long as it doesn't involve infringing on someone else, but the other person, the unborn child, also has the right to not be killed (and here it's important to make the distinction between the right to life and the right to not be killed). So, as there is potential for a conflict here between the rights of the mother and the rights of the baby, to figure out who has the right to do what in this situation requires a bit of investigation.
If the mother voluntarily engaged in sexual activity that she knew had the potential to result in a pregnancy, then it is reasonable to conclude that she is responsible for putting that baby in a vulnerable situation, and an implicit contract then exists that she not violate the baby's right not to be killed while it is fully dependent on her.
Take the example of the hot air balloon. Person A invites Person B to take a ride in his balloon. Person B agrees, with the expectation that he will, at the end of the ride, return to the ground safely. Mid-flight, however, Person A decides he no longer wishes to have a passenger, and "evicts" Person B, sending him plummeting to his bloody death. Was Person A simply exercising his property right in his balloon? Obviously, an implicit contract existed between Persons A and B. Person B was to be put into a vulnerable situation, and Person A was obliged to ensure his safety to the best of his ability. Person A violated that contract, and so is in violation of the NAP.
If we apply this example to the pregnant woman and her child, the case becomes even clearer. The child had no opportunity to consent to being placed in a vulnerable situation. The woman, as well as the man with whom she engaged to bring this situation about, are fully responsible for the plight of the child. To kill, or "evict", the child would be a clearer violation of the NAP than in the hot air balloon example.
That's all well and good for a child brought about through consensual sex, one might say, but what about in the case of rape? The mother had no choice in the matter, and is herself the victim of aggression.
In the case of rape, the mother has not made an implicit contract with the baby. But the father has. It takes at least one voluntary actor for sex to take place, and in this case the father, the rapist, is the one who is responsible for the consequences of his actions. Since he cannot physically carry a baby, I believe that, besides compensation for pain and suffering to the woman, he should be made to meet all financial needs associated with the carrying, birthing, raising or adoption of the child. He made the choice to rape, and so must pay for all the consequences of his actions. But I don't believe that the woman, even in the case of rape, has the right to kill the child, an innocent bystander (and in so doing save the rapist from having to own up to all his responsibilities).
In libertarian legal theory, when a person is murdered who has people dependent upon him, such as a husband/father, the murderer must not only compensate the family for the fact of the murder itself, but also for the financial hardship that will occur as a result of the loss of the family provider. The murderer cannot bring the murdered man back to life, but he can make the family financially whole. I believe we have a mirror image here in the case of abortion: Neither the death of a human nor the creation of a new life can be truly undone. A dead man cannot be brought back to life, and a newly created life cannot be uncreated. It can only be killed. When a rapist's action result in a new life, no one has the right to "take it back" and end that innocent life. That's not justice, it's just a further violation of rights beyond the initial violation of the mother's rights. But the rapist can make the mother financially whole.
A woman normally has the right to do with her body as she wishes, but as a woman I recognize the fact of nature that I have the potential to be in a position in which my exercise of that right would be a violation of the rights of another innocent human. The fact that a woman was aggressed against, in the form of rape, does not give her the right to violate the rights of an innocent. It does, however, require that the rapist compensate her not only for the initial rape but for the ongoing consequence of the new life.
Some libertarians will respond to this with appeals to Murray Rothbard's "evictionism". I respect Rothbard a great deal, but on this I believe he is mistaken. To "evict" the child from the womb, even if you don't directly kill it, is in fact an act of aggression upon that child, just as it would be an act of aggression to "evict" someone from a hot air balloon. And if our basic assumption is that people have a right to not be killed, and that peoples' freedom to act is constrained by the rights of others not to be aggressed against, then I think it's clear that the libertarian position should be that a child in a womb is a human life, put in a vulnerable position through no fault of their own, and have the right to not be killed, no matter the conditions that brought him/her into that position.
The considerations above are concerned with matters of justice, and in a stateless society in which the NAP was widely regarded as the benchmark principle of justice those concerns would suffice. But in a world of convoluted ideas about law, human rights, and enforcement under a state, things get a bit more complicated.
I'm aware of the massive violations of privacy and general insanity that could occur if abortion were made outright illegal on a federal level in modern America. I believe getting the government out of the medical industry completely would be a big solution to that aspect of the problem, restoring privacy between doctors and patients. I don't think that banning abortion in today's society is necessarily the solution. Our attitude concerning it as a culture must change. If we ban it and people still want it, there will be a black market for it. That's one of the strongest arguments against banning abortion, and it makes sense.
My purpose above was to address the morality and justice of it. If hearts and minds are not changed, legislation won't fix the problem. I believe in working to change those hearts and minds. And in a free society, an AnCap world, law itself would work very differently. I believe the manifestation of a cultural rejection of abortion would result in a situation in which this would be dealt with largely within families, as a very private matter, and possibly be addressed within decentralized, contractual law that reflected the morals and norms of those who voluntarily consented to the terms. The truth is, abortion is a difficult sort of murder to prosecute without massive violations of medical privacy, and vast power being granted to some entity. Families and communities are much better able to deal with the issue peacefully and without the baggage that comes with government legislation.
Some people might say that's a cop-out, proof that I don't 'actually' believe it's murder, but I'm regarding it as similar to when someone goes around murdering homeless people. Yes, it's murder, and it's horrible and the murderer must be stopped. But the homeless victim may have no advocates on his side; no one to notice that he's gone, or to investigate his disappearance. Do we then track each human being on earth to ensure that each murder is noticed and investigated? This is just reality. It doesn't make the murder any less heinous, it just means that the reality is that no one was there to fight for the victim. We're not going to get utopia and stop every murder, and an attempt to do so by an all-powerful state would give us hell on earth. Similarly with abortion, we can't/shouldn't track all women and give them monthly pee tests so we can record each pregnancy and investigate each miscarriage. Nor should we surveil each doctor's office, violating the privacy of all to catch a few. All this means is that the solution is not a statist one, it is a human one. We can save more lives by caring more about life than we can by giving power to the guns of the state.
As libertarians, we are often attacked for supposedly being "heartless" and not caring about the least fortunate in society. Of course, we know that it is precisely because we care about human life, dignity, and justice that we believe as we do. We know that committing an injustice in an attempt to rectify another injustice is wrongheaded and disastrous. We know that adhering to first principles is of utmost importance for forming a coherent worldview and ethic. I believe that the abortion issue has been a point of confusion among libertarians. Being pro-abortion has been given the label "pro-choice", and that has led many to conclude that being pro-life necessarily means that one is "anti-choice". This is a misuse of terms. It is not anti-choice to oppose murder, whether we're speaking of adults or children, born or unborn. It is simply a consistent application of the core libertarian opposition to the initiation of aggression. I hope, and I believe, that one day mankind will view abortion in the same way that we now view slavery: as an obvious and self-evident injustice. I hope that libertarians will be at the forefront of that ideological shift.