When registering a domain, the best place to start is either Yahoo or Google. Why, you ask. Well if you want anyone to find your site, you are going to need to be seen on Yahoo and/or Google.
Search for content that will be similar to your site, then see what other domains people are using for their sites. Odds are the domains listed at the top of Yahoo or Google have been quite popular.
Make sure that the domain name you choose makes sense to someone looking for you. If you are a dog groomer, a domain name like jennysdoggrooming.com is much better then k9dirtremoval.com. It should also be easy to remember, and the shorter the better.
You may also wish to purchase a few domains for your site. Such as jennysdogcare.com or cleandogs.com.
There are many different registers out there. My favorite is GoDaddy.com. There are very reasonably priced. As of this writing, it is only $9/year for a domain, with additional discounts for multiple registrations.
Once you register your domain name the next step is finding a hosting company, but that is another article.