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.
Names given to the Spanish language
Conversely, some nationalist circles prefer the term "español" because they perceive their ethnic community to be distinct from that of Spain and therefore do not object to the language of Spain being called Spanish. In Basque-speaking regions, whose language is not of Romance origin (Basque is considered by many scholars to be a language isolate), some Basque speakers also use the term "erdara" or "erdera" ("foreign") specifically for Spanish, since for them, it is the prevalent foreign language, just as in the French Basque Country, "French language" is the usual meaning of "erdara"; in fact, "erdara" is another language, different from Basque.