The legal name of a business is the official name of the person or entity that owns a business. If you are the only owner of your business, then its legal name is simply your full name.
If your business is a general partnership, and you have a written partnership agreement that gives a name to the partnership, then that name is the legal name of the business. Otherwise, the legal name of a general partnership consists of the last names of the owners.
Your business's legal name will be required on all government forms and applications, and is particularly important to use on your application for a federal employer identification number.
If you plan to use a name that's different from your business's legal name, you'll need to register the name you want to use with a government agency. For more information, see the following question, What is a fictitious business name?