In general, you cannot copyright a domain name, though there are other forms of protection that are extended to domain names and domain names associated with works protected by copyright. The laws regarding how domain names and copyrights interact are fairly new, however, so there may eventually be changes to how these policies have been established. Domain names are typically protected by other systems, and in general a person can own a domain name but not in the same way that a copyrighted work is owned and protected. While you cannot copyright a domain name, you can still copyright the contents of an Internet website.
A copyright protects an original artistic creation or work of art, such as a painting, written story, or computer software. Since a domain name is not an artistic creation, you cannot copyright a domain name, as it falls under a different form of intellectual property law. Similarly, you cannot copyright the name of a character in a story or even the name of the story or a book, band, song, or other artistic work. Copyrights only protect artistic works and creations, though you can potentially trademark a particular name for certain uses to afford some legal protections for use in business.
You cannot copyright a domain name because the domain name of a website is not considered an artistic work. A domain name can be registered with other agencies, however, to grant certain protections to the domain name. You could also potentially battle someone else over a domain name if its use infringes upon a copyright you own, such as a domain name that clearly refers to work that you own through copyright. This can be very difficult to prove, however, and it is often easier to prove infringement of a trademark rather than a copyright with regard to domain names.
The work that appears on a website can be protected by copyright. This means that if you create a website or blog onto which you post artistic works that you create, you own the copyright on the original works that are displayed on your website. Since you cannot copyright a domain name, you also cannot register a domain name for copyright protection with a government agency. The Internet Corporation for Assigned Names and Numbers (ICANN), however, provides registration for domain names and similar Internet identifiers, and you may consider registration with ICANN for domain name protection.