Yes, you can create a forum or a website on your name and you can copyright it just for you. You need to choose a good cyber laws acquainted lawyer in order to create your copyright statement and place it on your site.
Ignore the above post. Protecting the name/mark used in your domain has nothing to with a copyright, but rather a trademark. You can find a wealth of information on the web about the process, but you can't trademark a name that is already being used for the same or similar usage, and certain descriptive marks are not eligible. Whether or not you can have trademark rights depends on the actual name.