When anyone, whether an individual who just wants to self-publish his own poetry or a huge corporation like Wal-Mart who wants to sell their image registers a domain name, it immediately goes on record with ICANN (Internet Corporation for Assigned Names and Numbers), a complex, private sector, nonprofit company that has policed internet domains since 1998.
How do you find the records they keep? You can find it all online.
In reality there is no official WHOIS client, but Godaddy.com is a good place to start. Almost any web site that hosts domain names has a "whois" lookup client that you can use to find out who owns any particular domain name, or to check whether a domain name you want is available. If it isn't available because someone else has it registered, you'll be presented with a list of suggested alternate domain names based on the name you entered.
Alternatively, you can go directly to a site called Whois.net and type in the name of the domain in which you're interested. If you're trying to find out who owns an existing domain that information will be there, usually below the search results. You'll be presented with contact information, including a physical address and the date of registration so you will know when the domain name expires.
Whichever you use, pay attention to the domain address. Widgets.com is a different site from widgets.org and still different from widgets.edu. With several more addresses added in the last few years, and addresses coming available which denote various countries, you need to know exactly what you're looking for.
Most domains are public information, although there is a security feature available that allows third parties to mask the owner's information by providing their own in its place. For instance, Network Solutions advertises that they will, for a fee, hide certain private details from the public. They do this by substituting their own information on the registration form so that everything points to them. That means it isn't always as easy as it may seem to find out who really owns a domain name.
If you have a very good reason for needing to find an owner who has found it worthwhile to have their information blocked in this manner, sending them an email or calling them may help if you have a valid need. Even then, you may not be able to directly contact the domain owner.