Names given to the Spanish language
Formally speaking, the national language of Spain, the official Spanish language, is the Castilian language (as opposed to regional Spanish languages like Galician, Catalan, Asturleonese and Basque). As such both names, "español" and "castellano", have distinct and independent meanings that may be required for clarity in some specific contexts.
Generally speaking, though, both terms can be used to refer to the (national) Spanish language as a whole, with a preference for one over the other that depends on the context or the speaker's origin. "Castellano" (as well as "Castilian" in English) has another, more restricted, meaning, relating either to the old Romance language spoken in the Kingdom of Castile in the Middle Ages, predecessor of the modern Spanish language, or to the variety of Spanish nowadays spoken in the historical region of Castile, in central Spain.
Burgos
Spanish, also called Castilian, originated in Castile and spread throughout the Crown of Castile's possessions following the reconquista. It was eventually dispersed throughout the Spanish empire following the conquest and colonization of the Americas, and Spain after the union of the Catholic Monarchs's kingdoms being co-spoken with other languages. Of the many different variations and dialects found throughout the world, it's argued that the purest Spanish can be found in the region of Burgos due to the fact that the language first developed here in the 10th-century.