Getting a domain name involves registering the name you want with an organisation called ICANN through a domain name registrar. For example, if you choose a name like "example.com", you will have to go to a registrar, pay a registration fee that costs around US$10 to US$35 for that name. That will give you the right to the name for a year, and you will have to renew it annually for (usually) the same amount per annum.
If you have a web site, you should seriously consider registering your own domain name. A domain name is a name like "thesitewizard.com" or "thefreecountry.com", which you can use to refer to your website. Note that you do not have to be a company or organisation ("organization" if you use a different variant of English) to register a domain name. Any individual can do it too.
Namecheap is a solid choice for registering your domain name. Plus, their site is incredibly intuitive and easy to use, especially on their domain management pages, which can be incredibly helpful. They offer reasonable priced domains, and have a free DNS service, and WHOIS protection. They also offer SSL encryption, for those looking to beef up the security of their domains.