Spanish personal names are very particular and different from English or French names for example. My full name is "Jaime Bonet García". The breakdown of this is one first name (Jaime) plus two family names (Bonet and García). Let me introduce a bit of my family to you so I can best explain how this works:

My parents' names are: José Luis Bonet Izquierdo and Raquel García Linares

Hispanic women don't change their family names when they get married. My mother's name was "Raquel García Linares" before and after getting married. The children of "José Luis Bonet Izquierdo" and "Raquel García Linares" will carry the first family name of each parent. Hence, my full name is "Jaime Bonet García". My father in turn got his first family name from his father and his second family name from his mother:

My father's parents are: Tomás Bonet Moreno and Dolores Izquierdo Casal.

That's why my father's full name is José Luis (a composed first name here to further complicate things) Bonet Izquierdo. So every Spanish or Hispanic person has two family names—the first family name from his/her father plus the first family name from his/her mother.

For practical purposes, when dealing with English or French persons I introduce myself simply as Jaime Bonet. Actually, this is what I would be called even in Spain when talking in a simplified way, and it avoids me being addressed as "Jaime B. García", "Jaime García Bonet" or creating any other misunderstanding because of this difference. But now you know better!