No, copyright does not apply to domain names.
When domain names are registered the owners thereof have rights to the usage of the domain names. Even if someone uses the exact same name, but with a different extension you will not be able to do something about.
If your domain is called example.com and someone else registers example.net that person will have the same rights to the domain.
Trademarks however, apply to domain names, but only if the owners thereof have applied for trademark rights to those domain names.
Specific conditions for trade marking must be met before a domain will be able to have the exclusive protection afforded by a trademark.
If you have branded your company name or products through a trademark and wish to extend the trademark to the domain, you will have to apply for it.
The best route is to contact Intellectual Property lawyers to handle the process on your behalf.