Property, in its common conceptualization, is simply an item, process, idea, or region of space to which one lays claim and society acknowledges said claim. Usually this claim comes in the form of a title, deed, or bill of sale (receipt) issued by the seller or government and the acknowledgment takes the form of deference to property and licensure laws (and all the violence inherent therein). Of course, an anarchist would be disinclined to use such a convention. Before throwing it out, though, we ought to explore the function of such a convention.
What function does property convention serve in contemporary culture? Ostensibly, it is a determining factor in the acknowledgment of rights and liabilities. If some thing is one's property, one has the right to do with it what one will; if said thing is not one's property, one can only do with it that which is permitted by the acknowledged owner. Of course, if this rhetoric is to be used as a definition of property, the state is the owner of all property by virtue of laws concerning what uses are allowed for all property. As a function, though, this idea serves as an excellent starting place for defining property.
What could be necessary and sufficient conditions for an item to be considered property, with an eye to producing a functionality similar to that mentioned above? I am not confident that I could fully formulate them here, but I must at least begin the discussion.
One such condition I believe to necessary is the discrete nature of the object in question. This is both a practical and a definitional concern. Where one could indicate a clearly defined object and lay claim to that specific object, it would be nonsense to indicate something nebulous or unconstrained and lay claim to it. For example, I could lay claim to a specific apple as my property and the claim could be easily constrained to the perceptible material boundary of the apple's flesh, but it would be practically and conceptually impossible for me to crop-dust the line at the DMV and then lay claim to the fart cloud permeating the room and everyone's nostrils. Another possible example would be for me to fence-in a distinct and limited area of land and lay claim to it and it may be a legitimate claim. Barring that being a legitimate claim, I can at least claim that the fence is my property.... which, given the basics of property rights, would be more or less functionally equivalent to claiming the land inside the fence.
However, laying claim to “all the land between this ocean and the next” or “from sea to shining sea” would be as absurd as, and logically equivalent to, claiming “all the apples”. It may be possible that, by way of trade (purchase) or labor, I may successfully acquire all of the known apples or all of the known land, but this is a circumstantial quality of that claim, not an essential or necessary one. What this means is that an apple or piece of land could be discovered to which I have not yet laid legitimate claim and I would then no longer own all of the apples or land, whereas, a categorical claim to “all the things” would mean that if a new “thing” were discovered, it would automatically be incorporated into my property.
In addition to being a discrete object, one must claim it in order for it to become property. A rock in a forest, unseen by man, may as well be a planet in the Andromeda galaxy: undiscovered, unowned, unimportant. A rock in my hand, though, is mine (unless the rightful owner has handed it to me... whatever). Of course, if two parties lay claim to an object, there must be some principle by which to determine whose claim is legitimate, but that problem should be explored later.
There are those who claim that whatever object is claimed, it must be a limited resource in order to become property. I hear AnCaps assent to that claim, but only because they believe it to be a tautology; all resources are scarce, only the degree to which they are scarce is in question. I also hear liberals and liberal-leaning individuals assenting to this claim. However, I believe this claim is assented to solely because it can serve the liberal agenda of eliminating property; in a post-scarcity world, property couldn't exist. I am inclined to agree that such a condition for property is tautological, but for a different reason than the AnCaps. In laying claim to portion of even a non-scarce resource (such as a bucket of seawater or a fistful of sand at a beach), one creates a particular type of scarcity. The sand owned by the individual in question is scarce by virtue of being his sand. It can be argued that such a distinction is meaningless, that his sand is indistinguishable from any other which is non-scarce... but the industry of religious and cultural relics would argue otherwise. Of course, this scarcity only exists so long as one cannot lay claim to the entirety of a resource, scarce or not, as I addressed above.
The next condition is complex and may be considered the deciding factor by which property is defined. Property could be considered an object to which one has the most full control over and access to. While this principle sounds straightforward and implies a Hobbesian brutality, it is far more nuanced and just than it sounds. Some examples really are simple and Hobbesian. If I wander into the woods and find our previously undiscovered rock from before, I can pick it up and claim it as my own. It is mine by virtue of being in my hand. If I bring it home and place it in my yard or home, it is mine by virtue of being on my property and in my control. If I were to claim it as mine, drop it in the woods where I found it and walk away, never to return again, anyone could claim the rock because I have forsaken control over and access to said rock.
This is where simplicity breaks down. What happens if I loan said rock to my neighbor, or rent it to a client? I abdicate physical control and access to someone else, how is it different from simply having gifted or sold it to someone else? An easy answer, which I reject, is “contracts”. Contracts, as readers of this blog will know, are as fictitious as the law in my mind. A lease or renter's contract amounts to nothing more than a promise with a written reminder of that promise. Which, incidentally, serves as an excellent launching point for my suggested solution. So long as my neighbor honors my wishes regarding the rock (contractually state or not), I still exert control over the rock, if indirectly. I may lack access to said rock, depending on the nature of my agreement with my neighbor, but insofar as the nature of that lack is temporary, my control of the rock serves as a substitute.
If my neighbor decides he values my rock more than his friendship with me, he may decide to violate our agreement and claim the rock as his own. In which case, he takes control of the rock and renders my lack of access to it permanent. These two factors being the determining factors of property would render the rock as his property. At this point, the Leviathan of social acknowledgment reawakens and rears it's many ugly heads. One method by which I could prevent or mitigate such an event would be to publicly establish my claim to the rock before lending it to my neighbor. At which point, if my neighbor steals the rock, I can call upon our community to place social pressure on him to return my property. I could also alert the community that I intend to retrieve my property at any cost, in order to dissuade the virtuous men of my community from interceding on my neighbor's behalf if such an attempt were to escalate to violence. In this way, the social forces I can bring to bear could still be considered “control”.
There are many technologies that lend themselves to this solution, though. Branding, titles/deeds, third-party records-keepers, smart property, remote bricking, social conventions, etc. have all come from the long-standing intuition of social acknowledgment. Even with such technologies, as a principled anarchist, I believe the utility of such solutions may be insufficient justification in light of the potential for abuse. One can turn to fictitious accounts of the wild west for examples of abuse of the contract/social acknowledgment solution to theft. These flaws are sufficient to encourage me to look for a better resolution, but this is one stopgap measure that seems to parallel the average person's intuition and lend itself to marketability. Other options have been explored and are in use today. I believe that a small enough community would have no need for such technologies, especially a community centered on anarchist principles. In the case of larger communities, I recommend reading Spooner, Rothbard, or (oddly enough) Proudhon for alternatives. Again, I believe in letting a thousand flowers bloom and seeing what works, but this particular issue and resolution is raised this time for definitional purposes.
The issue of theft gives rise to the last condition I have found to be necessary for a thing to be considered property; legitimacy of claim. My neighbor claiming the rock he borrowed from me as his own is considered an illegitimate claim and, therefore, theft. There is an overabundance of issues and theories concerning legitimacy of claim. Most straightforward of these issues and solutions is that of homesteading. It's a very involved discussion that I will explore further at a later date. A quick overview, though, is essentially thus: the first person to use or add value to an unowned object has a legitimate claim of ownership. After one establishes such a claim, one may give or sell it to whomever one so desires.
By extension, then, if one purchases property from someone else, one then has a legitimate claim to it. Of course, if the seller has stolen said property, there remains the issue of whether the original owner's claim or that of the unwitting purchaser is (more) legitimate. Thus far, I have not read or synthesized a categorical answer to this particular problem. Perhaps discussions with you, the readers, will help me to do so. In the mean time, I'm inclined to give the pragmatic case: don't let your stuff get stolen. If it does get stolen, get it back before the thief sells it. If you fail to do so, there's little cause to steal it from someone uninvolved in the crime. Maybe, if you ask nicely, you and the new owner can work something out.
This answer is clearly unsatisfying, so I will venture two cases that each have their own flaws, but may prompt appropriate discussion. If ownership is some metaphysical attribute that a person or piece of property has, then the claim of ownership that the original owner makes would be more legitimate than that of the thief or the unwitting purchaser of stolen goods, as theft would not transfer the metaphysical ownership from oneself to a thief... or we wouldn't call it theft and expect the thief to return the stolen property or equivalent goods. If, in fact, ownership is merely a term we use to describe a practical state of affairs, the purchaser of stolen goods has a legitimate claim to the property by virtue of having engaged in a voluntary exchange of goods, trading something of approximate value for something else of approximate value.
There also arises the issue of homesteading abandoned property, which closely parallels the limit cases of theft. I feel I've taken enough of your time this time around, and I believe any satisfying answers would have to be informed by the limit cases of theft and their resolutions, so we'll save that issue for later. For now, titles, records, blockchains, receipts, deeds, etc. are useful tools to try and avoid finding oneself embroiled in such an issue.
TL;DR; My working definition of “property” is, “any discrete object to which one has access, control over, and a legitimate claim by virtue of homestead or acquisition from the previous owner with the owner's assent.” This definition needs improvement and exploration, especially in establishing a principled resolution of theft and homesteading abandoned property.