In my work, I encounter this particular confusion quite regularly - clients (and adverse parties) seem to think that copyright applies to both a website and its domain name. But this belief is incorrect. Here's a simple, easy way to remember it:
Copyrights are for Content. Trademarks are for goods/services.
So, a client can claim a copyright in the content written on a blog, a website, etc. However, you can't claim a "copyright" in the domain name that hosts your website: it's not really content. It could be, however, a name linked to your goods or to a service that you provide. If so, you can claim trademark rights in that name.
[Note that these rights usually apply whether you register with the Federal government or not! But registration certainly helps your arguments later on, if it comes to that...]
For further information, the US Copyright Office website has a document that explains in greater detail what kind of things can get copyright protections, including literary, musical, and dramatic works. And, likewise, the US Patent and Trademark Office (usually abbreviated USPTO, or just PTO) has a document that explains trademarks in greater detail - it describes them as "a word, phrase, symbol, or design, or a combination thereof, that identifies and distinguishes the source of the goods [or services] of one party from those of others...." Each US state may also offer trademark or copyright registrations, typically at a lower filing cost, although the usefulness of local registrations may not be great as federal registrations.