Understanding Hispanic surnames of 2 last names

Although my roots are from Puerto Rico I grew up in New York. Like most other Hispanics living in the US, I only have ever had to use one last name. What's interesting is that most Hispanic countries use two last names. 

Why two? 

What's the point of having two last names, you wonder? 

Let me explain...

I learned that in Spanish a last name is not called a last name (último nombre). In Spanish, the last name has a name of its own, it is called an apellido. The proper translation to
English is surname. Surname (or apellido) does not mean "last." So, when you talk about someone's last name you talk about their apellidos (surnames) since there are two of them. The two surnames are referred as the first apellido and the second apellido.

Lets take my father's name for example. 
His parent's names are:
Father: Antonio Rodriguez
Mother: Andrea Pabellon 
His proper name in Puerto Rico would be:
Jose Rodriguez Pabellon

The father's last name is always the first surname, the surname that follows that would be the one of the mother. However, if the parents were not legally married then often times the child will only carry one surname & that would be the surname of his mother. 

So, what happens when you get married? 

Like in the US, nothing changes for the husband. However, the wife usually changes her name as follows. Her first surname remains the same (her father's first), but her second surname often changes to that of her husband. Sometimes including the word "de" is added between the two surnames to indicate that the second surname is her husband's. 

Here is an example of my great grandmother's name change during marriage. 
Her name before marriage: Petra Pabellon de los Santos
Father's surname: Pabellon
Mother's surname: de lost Santos

After marriage she dropped her mother's surname & added her husband's name. Her new name after marriage was Petra Pabellon de Ortiz

Now all of Petra's children's names from this marriage would carry the surname of Ortiz first then Pabellon would be their second surname. 

In the US, a family is addressed by the last name of the husband. In Hispanic circles, the family is addressed by the combination of the first surname of each of the partners in the marriage.

It was so confusing to me at first. Keeping all the surnames correct & in order was tricky but makes sense now. It's actually super helpful when trying to trace both of the parents to someone.