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.
Spaniards
Spaniards are a Romance ethnic group and nation. They are indigenous to Spain and share a common Spanish culture, history, ancestry, and language. Within Spain, there are a number of nationalisms and regionalisms, reflecting the country's complex history and diverse culture. Although the official language of Spain is commonly known as "Spanish", it is only one of the national languages of Spain, and is less ambiguously known as Castilian, a standard language based on the medieval romance speech of the early Kingdom of Castile in north-central Spain and the Mozarabic dialect of the Taifa of Toledo which was incorporated by the former in the 11th century. There are several commonly spoken regional languages, most notably Basque (a Paleohispanic language), Catalan and Galician (both Romance languages like Castilian). There are many populations outside Spain with ancestors who emigrated from Spain and who share a Hispanic culture; most notably in Hispanic America.