There are literally hundreds of websites where you could register a new domain name (URL). Finding a good Registrar is important. And most importantly knowing what you are looking for when choosing the right Domain Registration site for your needs is essential.
Here is a list of questions to ask yourself when considering who to use for buying your custom website and hosting:
Is 24 Hour Support important for you?
Maybe you are already pretty savvy with setting up your DNS configurations or your site forwarding. But if those terms are foreign to you then you will probably want to have a team of people who will be there to answer your questions. Whether by chat or by phone, it's a good idea to have support when you need it.
But remember you might pay for that in the cost of the domain. www.1and1.com has a great start up deal on some domains - starting $0.99 for the first year. Check them out see what you feel!
Are you wanting to SAVE as much as you can?
If having support isn't a big deal for you, and maybe you just want to buy a domain name because you like the real estate and maybe you want to sell it, or lease it, or build a site on it at a later date, then a site like namecheap.com could be your place with sites starting as low as $3.98!
Do you want a Do It Yourself Website Site Builder?
There are a few sites out there that are really great for choosing a theme of your liking and customizing it to your style and preferences. The most enjoyable, versatile and user-friendly platform that I have been able to find is called Weebly. I've been using Weebly to design sites for years. There is a lot of freedom and really awesome options for eCommerce sites, artist sites, and really great themes to choose from!