On the whole, we as humans tend to understand ownership and that it doesn't just come about in a vacuum. We like to own things because it not only makes for a good incentive to work hard, ownership is important for economies in that if you cannot own something you cannot trade it either. Libertarians tend to call owned goods property, but there are those who are anti-propertarian while maintaining there can be some semblance of ownership. Not only is this a misinterpretation of the concept of property, it happens to be a fallacy filled argument.
What is property? Well, P. J. Proudhon would tell you it's theft, and many people of the left-anarchist ideologies say it's legal fiction that allows "exploitation", but put simply, property is a right to ownership over an item or good.
Let's back that train up. I've read Proudhon's What is Property?, and while he says no less than a bajillion times that property is theft (even beginning the book with "WHAT IS PROPERTY? IT IS ROBBERY!"), he offers little logic behind the argument. I think the sentiment behind it is as von Mises thinks: That because modern societies have made into property things that were once taken and distributed by conquest at some point in history, we should abolish property and start again or at least redistribute ownerships for great justice.
It's a fine claim, and it comes up every now and again on the gateway to hell that is /r/politics, but in practice it can't possibly work. We can start with the United States of America conquering Native American tribes. Most of the victims are dead or have been for at least a few hundred years. Even if we opt to give to their families, from who are we giving things? Are we going to steal and un-ownership-ize people's holdings in New York, Washington D.C., Atlanta, etc. just so we can give it to a group that was wronged? I get that people need to make amends and recognize the evil that was done in history, but there is simply no just way a government (monopoly of force) can redistribute goods in that manner, let alone any manner.
Even if we get past the practicality argument, there's the economical one too. We are all standing on the shoulders of those who came before us. If we decide that it is justice to demolish a current creation because we must go back in time to make right for the past, we have some awkward problems to deal with. This would be like disbanding the oil industry every year or two, because it would be claiming that justice is making each oil company pay to rediscover and recreate distribution infrastructure all by themselves. This breaking down of capital would mean that no progress would be made. Just like we say it's fine that a new oil company can use the technology its predecessors have created, we should say it's fine that what was once conquest provisions can be made right if we just use it correctly. Life isn't about having a good hand, it's about playing a bad one well.
The legal fiction argument is a little off too. Proudhon and the left-anarchists usually make distinctions between possessions and property, possessions supposedly being natural ownership rights everyone respects, and property being some weird boogyman (giving a legal "Right to Increase" as Proudhon puts in his singular axiom.) that seems to incorporate every activity a modern business in a free market might do with freedom of contract (interest / usury, absentee ownership, and or anything else that sounds like it hurts someone). This "right to increase" is supposedly a special grant that allows people to unjustly manipulate their station by running roughshod over the rights of others. What makes it weird from an anarcho-capitalist and libertarian view is that libertarian law is not a bunch of laws that dictate who can use what in which ways, but rather it is a mode of dispute resolution. Property in a libertarian society isn't a set of rights given, it is a set of rights affirmed.
Everyone is free to claim ownership on anything they like, but if ownership is disputed then an arbitration case can be brought, where the arbitrator codifies society's norms about ownership by his personal judgement. As arbitrators are free market businesses, they must compete to provide the law that the most people want, and so most arbitrators would not turn property into an arbitrary behemoth like "intellectual property" (non-property because it is non-scarce), but would rather make clear the social norms in order to attract the most customers. As any anarcho-capitalist can tell you, if an arbitrator can't be fair enough to get two or more people to agree to use him, then he has no business. In effect, property is possession is ownership to a libertarian via market (libertarian) law because in order to avoid being taken to arbitration each person must respect property ownerships of others where possible (which is exactly the definition of possessions to a left-anarchist). The "legal fiction" argument just implies that the person has little idea of how law is supposed to work.
Now, what's important here is not how different the left and right anarchisms are, but how similar the two sides of the spectrum are. We like the concepts of ethics, justice, freedom, and order, it is just that we disagree on the issue of ownership and the law. Both sides of the line seem to like the idea of ownership just the same, and each would prefer not to let ownership get out of control. We only tend to differ on the necessity of law when it comes to ownership (or in some cases, the utility of a legal system at all). Needless to say, it gets really awkward at dinner parties.
What our (the libertarian) goal happens to be is to convince people that anti-propertarianism is illogical. The property-possession divide is really vague. The lefties like to hate on property for being terrible because it allows people to exert dominion over other people, but that's not what property is as a concept. Property is there to allow us to exert dominion over items and goods. We don't go out and say we own people, even if we have them under contract.
We might call a shovel a possession, let someone else borrow it, ask them not to break it, and when they try to we can step in and prevent that from happening. Similarly, we might call a shovel property, let someone else borrow it, ask them not to break it, and when they try to we can step in and prevent that from happening.
I can call my house my property, and if somebody breaks into it while I was gone and planned to return someday, I should be able to take the burglar to arbitration. Similarly, I can call my house a possession, and if somebody breaks into it while I was gone and planned to return someday, I should be able to take the burglar to arbitration.
Everytime I blur the line for one of these guys, I get a "you just don't understand, go develop an understanding of ownership" type response, when in fact the problem is that being anti-property is very much the same as being anti-ownership: If you can't claim absolute dominion over a good and have it justified by societal norms in a court of your peers, then do you really own it? If you can't truly own goods or even the ones you produce, are you really free? The end result is anti-property people arguing against property because it is a legal concept but somehow for ownership when ownership can only be codified through jurisprudence of common law.
So next time you get involved in a debate with anti-propertarians, go ahead and let them know they're using circular logic and special pleading with trying to differentiate property and possessions, and that all they need to do is look at it from a different angle. Ownership requires dispute resolution (law) to arbitrate between unclear claims, and in a free world, law (dispute resolution) can only codify and affirm social norms (like ownership).