One of the biggest objections to Anarcho-Capitalism is the belief that without Government, there can be no way to get recourse for a crime committed against you. This is untrue.
There's a lot of diversity within the Libertarian movement as to what kinds of law, law enforcement, courts, et cetera are best. My position is not necessarily the one held by most, or even any significant number of Anarcho-Capitalists or Libertarians.
So what would we do about the courts?
Let's start with the basis of justice, injustice.
1. The Crime
Adam lets Ben borrow his car. Adam asks for it back, but Ben does not return it. (I chose names beginning with A and B to help you remember who is who.)
2. The Judgement
Adam goes to a respected arbitrator and put his case before the court.
In a totally voluntary society it would be likely that written contracts would be used extensively when something as valuable as a car is at stake. So, for example, if Adam let Ben borrow his car, Adam might write a contract saying that "Adam is allowing Ben to use his car until Adam asks for it back". It would include Ben submitting in advance to any punishment the court (which may be named in the contract) decides on for breech of contract. Ben could make the case in court that Adam permanantly transferred ownership to Ben, that Adam gave him the car, not let him borrow it. If Ben had a contract with Adam's and his own signature which transferred permanant ownership of the car to Ben, and Adam did not have a later dated contract transferring ownership of the car back to him, Ben would not be held liable for returning the car to Adam, because as far as the contracts say, the car belonged to Ben.
b) Hard Evidence
Suppose Adam did not have a contract, however. By making some indelible, obvious, distinct mark upon the car, perhaps an unremovable license plate style identifier, perhaps with Adam's name and signature on it, welded onto the rear bumper, perhaps, and by claiming that a car with such identifier was now in the possession of Ben, Adam would be able to make a case. The court may ask Ben to bring the car in question to the court, to be allowed to make the case that it is not actually Adam's car that Ben has possession of, but a similar car belonging to someone other than Adam. Of course this would be totally voluntary on Ben's part, although refusal will only bring up suspicion and not help Ben's case.
It should be noted however that courts will be hesitant to judge such torts without contracts.
Adam would be allowed to bring in witnesses to testify that they saw Ben with Adam's car. Witnesses would have to sign a contract with the court saying that should their testimony be found to be intentionally deceptive, they would sumbit in advance to any force used by the courts to enforce retribution or reparation or whatever the court uses to convince witnesses to not give false testimony.
3. The Reparation
a) Under Contract
If the court (which was agreed upon in the original contract would therefore be entirely voluntary) decides that Ben has a car belonging to Adam, and that Ben needs to return Adam's car, Ben would then be obligated to return Adam's car. If Ben should then refuse to return Adam's car, Adam may go back to the court, and demand enforcement action, which would have been voluntarily submitted to in advance by the original contract. The court could then send it's enforcement agents to forcibly return the car to Adam. Ben would not be allowed to charge the court's enforcement agents with assault because Ben had already given them permission to assault him if necessary, the moment Ben signed the contract and accepted the terms of it. However, if Ben should resist and attack the enforcement agents, the agents would be able to bring charges against Ben in the same way Alex did.
b) Absence of Contract
If the court decides that Ben has a car belonging to Adam, it would then order Ben to return Adam's car. At this point, Ben could either get in touch with Adam and ask for a retrial in a different court that the two could agree upon (contractually such that it's decision would become obligatory and enforcement would become voluntarily submitted to in advance), or Ben could give Adam the car in question, or Ben could refuse to return the car and not demand a retrial.
If Ben decides to give Adam the car, that is the end of the dispute.
If Adam and Ben can agree upon a second court, the retrial will take place with a different and agreed-to arbitrator, whose decision becomes voluntarily obligatory and binding, the process repeating itself though under contract.
If Ben does not return the car, and Adam and Ben cannot agree on a new court for retrial, Adam returns to the court. Adam asks the court to declare Ben to not respect ownership, and thus not have rights (understanding and respect of ownership is prerequisite to having rights, and if any being can be shown not to respect ownership, it cannot be said to have rights).
Ben would then have his rights nullified until such time as Adam gets his car back. Any attempt by Ben to press charges against anyone else in court would be denied, as Ben would not be said to have any rights. Ben would become an "Outlaw," essentially an animal, until Adam was restituted and hence anybody would be free to kill, enslave, or take property from Ben without Ben having any legal recourse. Anyone Ben attempted to bring suit against could merely point out that Ben has been declared an outlaw.
4. Reinstitution of Rights
After Ben is declared an outlaw, he would obviously be in great danger. Adam could freely come and take all of Ben's property. So what could Ben do about it?
Well, Ben could go to the court which declared him an outlaw and attempt to get the ruling overturned. Obviously this would be the ideal. However that court has already decided that Ben has wronged Adam and not restituted him. If Ben believes the court to be crooked, he may go to a different court and sign a contract (which would make the second court's decision binding) asking for a restitution of his rights. The other court could then ask for the evidence from the first court. If the first court provides none, the second court could restitute Ben's rights. The two courts in disagreement would then contractually find a third court to find one or the other court to be biased and the decision thus invalid. If the two disagreeing courts did not do so, anyone not respecting Ben's rights would be charged in Ben's court, including enforcement agents from Adam's court. Depending which court the third court found to be errant, the decision would be binding upon both courts and Ben would be declared with finality to have rights or not have rights.
However, let us assume the second court recieves the evidence from the first court, to review the case. Ben would of course be allowed to review the evidence to ensure that nothing on his side has been left out. Then the second court, one of Ben's choosing, and not Adam's, would review the case. If the second court agreed with the first court, Ben would be declared an outlaw. If they disagreed, the third court would be sought as described above.
In the event Ben cannot recieve rights, Ben will have the option of entering protective custody. He could voluntarily enter a jail which, rather than keep outlaws in, would keep potential attackers out. He would be free to leave, though if he does so he risks being attacked without possible recourse. He could also transfer his property to the jail for protection, property to be returned once Adam has been repaid and Ben is thus no longer an outlaw. Should Ben do this, however, the jail which would then own all of Ben's property would be likely to restitute Adam. The Jail's revenue could come from Ben's work, or out of the property held in safekeeping for him, depending on the policy of the jail. When Ben's work surplus is sufficient that the jail can restitute Adam in money, if not in returning the car. At that point Ben would have his rights restituted by the first court, and when Ben left the jail, he would have his property returned to him, minus perhaps any payment that the jail took.
And so we have a free-market judicial process which is voluntary, simple, and contains within it checks to prevent corruption. It does not require a State to function, it focuses on restituting the victim rather than punishing the offender, satisfies everybody and oppresses nobody.