There are two companies that I recommend for purchasing domains from:
The first company is 1and1.com, and the reason I recommend them is because you can purchase a domain from them for only $8.99 a year, and they give you FREE private registrations! Click here to learn more!
The other company I recommend is www.GoDaddy.com. They have awesome tools that make it very easy for you to come up with a good domain name. They also offer excellent customer support if you ever have questions.
Please make sure to purchase a domain BEFORE you purchase web hosting, because you can save money by doing this, and you will also need a domain before you can set up web hosting.