You can't copyright a business name, but you can register is as a trademark or service mark (as Owen mentioned). This is done at the federal level.
When you incorporate your company, you (or your lawyer) are supposed to check the state's records to make sure that there are no other corporation with the exact same name registered in your state. So once you are registered and in good standing (paying annual taxes and fees), then technically no one can operate a business with the exact same name in your state of incorporation.
All the trademarks and corporation names are in the public records available on the internet.