The underlying framework of the Internet works much like regular old mail. To hold a conversation via mail, you write letters. Letters are just like "packets" on the Internet. Everything on the net gets sent around in these packets. Consider what a letter consists of: the actual letter (data), placed into an envelope ("headers" attached to the data), and a pair of addresses (source and destination ip addresses). You don't need to know how the letter will get where it's going, you just need to know the destination, and to put that in a standard location (the front of the envelope, equivalent to putting something in a fixed place in the packet header). Then you hand it off to the post office (your Internet Service Provider or ISP). You generally have a standard "first hop". For a letter, that's just your mailbox. It's the "default route" for your network packets. You usually put a return address on the envelope for two reasons. First, if the postal system has problems delivering your letter, they can inform you (usually by sending it back). Second, if the recipient for some reason needs to reply, the return address can let them know how the reply should be addressed. Notice that nothing requires you to put a valid return address on the envelope. All these things too are just like the Internet. (Technically, what I'm describing is the Internet Protocol, or IP).
This brings us to a problem. Computers really only understand numbers. Every computer attached to the Internet (with some exceptions where someone else acts on your behalf, kind of like having a P.O. box) has a unique address, called the ip address. For example, my main machine right now has ip address 126.96.36.199. Now we go back to the problem of serving people. Again, people don't do so well with numbers. So, the early network designers came up with a system. This system maps easy-to-remember names to numbers. www.hod4.net maps to 188.8.131.52. All your packets need to know they are going to 184.108.40.206 to talk to my server, but you don't have to type that in (I do, you don't). You just type in the host name (www.hod4.net) and a service translates this for you automatically. This system is called the domain name service. They called it that because the system they designed works hierarchically, with everything broken into domains and subdomains. For a long time the number of so-called "top level domains" was extremely limited, the now famous .com, .net, .edu, .gov and .org extensions you find still on most addresses. This is only now beginning to change (essentially because this gave certain groups a monopoly on all the services association with the DNS system, and they wanted to keep it). In earlier days the hierarchy was more strictly enforced, but you still see the effects today. For example, most universities have a subdomain (e.g. caltech.edu) within the .edu domain, because .edu was supposed to be reserved for domains used by groups involved in education. You can have many levels of subdomains, (e.g. chem.columbia.edu is a subdomain within the domain columbia.edu, set aside specifically for the chemistry department). Finally, to address a particular machine, you have the fully-qualified domain name (FQDN), like www.hod4.net or www.amazon.com. These names identify specific machines within the related subdomains (hod4.net, amazon.com).
All this layering gets handled by the DNS service. Every time you type in an address your machine hasn't seen recently, it has to do a DNS query to get the ip address. Let's see how this works. Take the example www.yahoo.com. By now you recognize that www is a machine in the domain yahoo, which is a subdomain of the top level .com domain. So, behind the scenes, effectively what the DNS system does is work from the top down and follow the chain. First it asks an authority about the .com domain who the authority is on the yahoo subdomain (these authorities are called name servers, because they serve up name information within the DNS hierarchy). Finally, it asks the authority on yahoo.com for the ip address of www and voila, you have your ip address. You might observe that I skipped over how you find a name server that's an authority on the .com domain. This is another part of the present day monopoly. There are a set of "root" servers that your DNS system just has to know about that are authorities for all the top level domains. The system only works well because right now everyone uses the same root servers, which they mostly have to in order for the system to work well, because it only works well if everyone uses the same root servers, ....
So, all this brings us to the true meaning of "owning a domain". A domain name like hod4.net is just that, a domain name. It has no intrinsic meaning or value on it's own, just like I can make up any old street address I want. But just like you can't send a letter to someone until you have a way of knowing their address, until a domain has entries in the DNS system, the domain serves no purpose. However, in the case of domain names, people are now willing to pay to register a domain name simply for the perceived potential value of having rights to control that domain (owning the rights to coke.com would be worth many $$. It's an interesting point that the monopoly that runs the show came up with an ad hoc set of rules to decide who really has the right to a domain and who is just a squatter trying to capitalize off of having grabbed up the rights to a domain first. Even if you could get coke.com, at this point you probably could not hold on to it.).