Whose name goes first in a Spanish last name?

This is based on typical Spanish naming customs - there may be variations by country or family based on local tradition.

The father's surname goes first. So María, daughter of Juan García López and Marisol Alvárez Marcos would be María García Alvárez. Note that it's the surname of the mother's paternal side (Alvárez), and not the mother's maternal side (Marcos) that is passed on from the mother.

At marriage María would affix her husband's paternal surname to the end of her name, preceeded by de (which means of). For example, if she married Luis Castillo Arroyo, she would call herself María García Alvárez de Castillo.