If you already have a web host, obtain from them the names of their primary and secondary nameservers. Don't worry if you don't understand what these things mean. Just save the information somewhere. The information can usually be obtained from their FAQs or other documentation on their site, usually under a category like "domain name" or "DNS" or "domain name transfer" and the like. If you can't find it, email them. You'll need the information to point your domain name to your website after you buy your domain. Having said that, if you don't have a web host yet, don't worry. Just read on.
When you type google.com into your browser, the domain is split into parts by the “dot”, from right to left. The first part is “com”, so your computer checks the root zone file to see which registry manages the .com TLD. It then checks with this registry to see where the next part, “google”, points to. The registry returns the IP address 220.127.116.11, and your computer can then ask Google to send their homepage over.
ICANN renewed Verisign’s .com contract, this time with an interesting new clause — the $6 cap previously placed on Verisign’s fee was removed, granting permission for Verisign to raise their fee by 7% each year from 2007 to 2010, bringing it to $7.85. The justification given by ICANN was “to allow market forces to determine prices”, which I don’t quite follow, but I’m no Keynes. They also point out, in subtler language, that it’s the US government’s job to worry about Verisign’s monopoly, which is, I guess, fair.
If you have a web site, you should seriously consider registering your own domain name. A domain name is a name like "thesitewizard.com" or "thefreecountry.com", which you can use to refer to your website. Note that you do not have to be a company or organisation ("organization" if you use a different variant of English) to register a domain name. Any individual can do it too.
Most people assume a domain name ends in .com so if you buy a domain name with one of the other extensions (.net, .info, .org, etc.), you’ll have some extra work to get people to remember that your site has a different extension. Don’t automatically assume you should only buy domains with .com, though. Many sites have done quite well with other extensions. (Look at us!)
In order to register any domain, you must provide certain personal information that you may not wish to be publically searchable in the Whois database. Signing up for WhoisGuard is a great way to keep your registration data private. WhoisGuard acts as a "shield" for your searchable information, displaying the address, phone number, and email of the domain registrar (Namecheap, e.g.) instead of your own. Keep in mind that WhoisGuard is an optional security add-on that must be renewed separately when you renew your domain.