What is a domain name? A domain name is, well, a name. Something like radionet.com, phoneboy.com, or human.com. Domains, along with something called DNS allows the Internet to work as it does. An organization connected to the Internet will often have a domain name similar to the name of their company or relates to some part of their business. A domain basically allows people to easily find you on the Internet whether it be by email, by web, or by some other means.
Why get a domain instead of just using your ISP's domain for your correspondence and/or webpages? Obviously, not everyone needs their own domain, but for content providers, internet companies, people doing business on the Internet, or for those who want the prestige of having their own domain name, having your own domain has the following advantages:
Before you attempt to register a domain, though, you should check to see if the domain is already registered to someone else. Use should check the Whois database to see who, if anyone, has the domain you want. You also just can't register a domain unless you have someone who is willing to serve it for you and set stuff up so the domain can be used. You will need to talk to your ISP to see if they would be willing to serve the domain for you and set it up so your webpages can be used with this domain and email sent to your domain will end up in the right place. There may be an extra charge for this, so ask before you commit.
After you've done that, you will need to go through the process of registering your domain. The InterNIC's website has plenty of information on registering a domain, including a procedure document entitled "Registering a New Domain Name." You will need to work with your ISP in order to get all the necessary information to put in this form and submit it to the InterNIC. Your ISP may be willing to do this for you, but make sure you register yourself as both the administrative and the billing contact so that you can make modifications to your domain record without having to go through your ISP.
There are actually other domain registries out there and other ways to get your own domain, as it were. The AlterNIC allows for other sorts of top-level domains (e.g. xxx, web), but the AlterNIC is not supported by everyone, so use these domains at your own risk. If you live in the United States, you can get your own city.state.us-type domain from The US Domain Registration Service. The cost for the domain will vary depending on whom you are and where you live, but will work everywhere. For other localities, check out http://www.domain-registration.net/, which has information about other regional registries.
You might also want to check out Monolith, a free service that gives you a domain under ml.org. Aside from giving your website a custom ml.org domain, they also have a service that allows you to have a fixed hostname that follows your IP address, even if your ISP gives you a different IP address every time you sign onto the Internet!
One word of caution -- be wary of using third parties to register domain
names. They often charge you more than it would cost if you were to do
it yourself. Where possible, do it yourself. If you need help, talk to
your ISP or some other knowledable geek and they can get you in the right
direction.