Basically, a domain name is an easy-to-remember name, made up of letters and/or numbers, that points to a particular IP address. The gadget that translates domain names into IP addresses is called a Domain Name Server, or DNS.

A typical domain name comes in two parts, separated by a dot. The part to the right of the dot is called the top-level domain, and you must choose a top-level domain according to what country you're in, and what kind of organization you have. Most other countries have a countrywide domain, for example: 

.de is Germany (Deutschland) 
.fr is France 
.ru is Russia 
.ch is Switzerland

The United States has several top-level domains (many non-US sites use some of these domains too): 

.com is for commercial sites.
.net is for hosting services and other entities having to do with the Internet. 
.org is for non-profit organizations. 
.edu is for colleges and universities 
.gov is for government agencies 
.mil is for the military. 

Within each top-level domain, there is obviously a finite number of possible domain names. To use a certain domain name, you must register it with a domain registrar. Domain names are awarded on a first-come, first-served basis. Since the Web gold rush has been going on for a few years now, all the neat names have been long gone. In fact, almost all the possible combinations of 2, 3 or 4 letters have been snapped up - you can sometimes tell how long as site has been on the Web by how long their domain name is. 

Do you need to have your own domain name to have a Web site? Of course not. Your home page can be located in a subdirectory under your hosting service's domain, for example: 

http://www.yourisp.com/~yourname/ 

If you have a business Web site, however, having your own domain is highly recommended for the credibility it gives. A company without its own domain is perceived as a small-time outfit, or at least one that is not Web-savvy. Having your own domain costs little, so make choosing and registering a good domain name one of your first projects. 

Even if you have only a rudimentary Web site or none at all, it may be worth registering a domain, so that you can use it in your email address. Yourname@yourcompany.com makes you look like a legitimate concern, like having your own letterhead. Whatever you do, don't use one of the "free" email services or an online service like AOL as your business email address. A business that uses an email address ending in hotmail.com or aol.com will be dismissed as at best clueless about the Web, and at worst a fly-by-night outfit. 

Does this mean you have to ditch your existing email account, and set up a new one with your Web hosting service? Not at all. An email address that uses your own domain name can be set up to forward mail to any address you please. This is called an email alias, and is a valuable tool. Your hosting service can set up as many email aliases as you like, and point each at the address of your choice (don't let them charge you a lot of money for doing this - it involves about 10 seconds of work to set up an email alias). This can be useful for several reasons. If you change email providers, you can keep the same email address - just change the alias. An email address can also be associated with a role instead of an individual - for example, webmaster@yourcompany.com can be set up to forward mail to your address. If you later hire someone else as Webmaster, you can change the alias to go to their address. An alias can even send mail to several people at once - for example, staff@yourcompany.com could be set to forward mail to everyone in the company. 

Many small companies let a hosting service register the domain name for them, with the hosting service as the owners of the company's domain name. Such practice is strongly advise against. If you do it this way, the domain name will legally belong to the hosting service, not to you. If you're absolutely positive that you will be remaining with the same hosting service for the rest of eternity, then this will probably never be a problem. If you ever want to transfer the domain name to yourself, however, you'll have to do a Registrant Name Change Agreement, which may involve signing, notarizing, mailing and faxing a bunch of forms. If your hosting service never gets around to doing it, or decides they want to charge you a fat fee for doing it, there'll be nothing you can do.