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.
Castilians
Castilian (or Spanish) is the dominant language in Spain, and therefore was the language that was imposed in the New World by the Conquistadores during the Spanish colonization of the Americas. Due to this gradual process, the Hispanophone world was created. As Castilian was the language of the Crown, it became the official language of all Spain, used side by side with other languages in their regions for centuries. During the years of the Francoist State (1939 to 1975) there was an attempt to suppress the regional languages in favour of Castilian as the sole official language, causing a backlash against the use of Castilian in some regions after his death.