You need to find a domain registrar (someone who hooks you up with a domain name). You generally pay them less than $10/year for registration and renewals.
I use godaddy. I just purchased a .info domain name from them the other day for $.99 to post pics of my laptop for sale. Cheap and works great.
However, you don't need a domain name to have a site. You could just use your IP or a free subdomain that points to your IP. I use No-IP for this (
http://www.no-ip.com).
If you are hosting at home and use a domain, you will need a DNS and a dynamic DNS updater. My favorites are ZoneEdit.com and DirectUpdate.net.
Good luck on your site!