In the US, most people have three names, first, middle, and last. Some have more; some have less. Your legal name is the full name as it appears on your birth cirtificate. I think some churches may baptize you under a different name.
As people mature, they acquire the habit of using a particular form: W.E.B. Dubois, George H.W. Bush, John Smith, W.Somerset Maugham, E.E. Cummings, Ben E. King.
This is usually the way they sign their checks.
When you fill out a legal form, there are often blanks which are labled for your name:
last name____________ first name______________ middle initial_____
I think your "last name", or family name, is called your surname.
Your "first name" is called your given name.
Of course you can have your name legally changed, and that becomes your new legal name.