The Spanish verb llamar means to call. When conjugated for the first person singular, it becomes llamo. Therefore, when someone says “Me llamo…”, they are saying “My name is…”
The word nombre means name. Thus, when a person says “Mi nombre es…”, they are introducing themselves with “My name is…”
The purpose of both phrases is the same – to inform the person on the other of the communication of one’s name. The usage of either phrase is correct.
While both are commonly used, mi nombre es is taken to be more formal. Me llamo is conversational, while mi nombre es is regarded as more appropriate for written communication.
A third phrase – Soy or Yo soy – can also be used for introductions. It is the least formal. It is the first person conjugation of the verb ser – to be. It translates to I am.