Yes, I will agree you could set up your Website with the free server space that is offered at Yahoo!, MSN, Geocities, etc. This option is a great option if all you are interested in is creating a site that your family and friends can visit to see what you are up to or to see pictures of how your kids have grown. But for an online business or informational site here's why you should seriously consider Registering a Domain Name:

You will have total control over the look,feel and functionality of your website and it's design.
You will never have to worry about your website being shut down.
Your website address will be simply www.yourcompany.com instead of www.somefreeserviceprovider.com/communities/
yourscreenname/yourpages.htm
You will have a better chance of a better ranking in the major search engines and directories due to your domain name using words that relate to your company instead of someone else's.
I think you will agree that it is much easier for people to remember www.yourcompany.com.
Your email address will be yourname@yourcompany.com.
With your own domain name it gives your company a professional look and feel.

Registering a Domain Name is a very simple process...The most important thing I can say is Print out all your receipts and account information during the process. And be sure you write down all your username and password information and keep it in a safe place.

Here are my recommended Domain Registration services:

NUMBER 1 PICK!!! 1&1 Hosting Visit 1&1 - Best hosting solution along with one of the cheapest domains.

Network Solutions - One of the biggest with many options including hosting. Take me there!

Hostway - The best hosting anywhere and they also will register your domain name. Great solution for those that like the one-stop shopping experience. Take me there!

Domain Zoo- Offering Domains at a great price of $14.95/year. Domain services include Domain Forwarding, Domain Appraisals and Web Hosting Packages. Take me there!

Dotster - Currently has a transfer special for $8.95 and they add a free year to registration. They also offer Domain Registration for $14.95 a year, but also have acces to a sitebuilder and webhosting. Take me there!

Since you are here, I should mention two other great ways to get your domain name registered. And they also happen to have Hosting Solutions as well as Website Design solutions.
Read my article on Site2You!
Read my article on SiteBuildit!