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
Spanish was exported to the Americas due to over three centuries of Spanish colonial rule starting with the arrival of Christopher Columbus to Santo Domingo in 1492. Spanish is spoken natively by over 400 million people and spans across most countries of the Americas; from the Southwestern United States in North America down to Tierra del Fuego, the southernmost region of South America in Chile and Argentina. A variety of the language, known as Judaeo-Spanish or Ladino (or Haketia in Morocco), is still spoken by descendants of Sephardim (Spanish and Portuguese Jews) who fled Spain following a decree of expulsion of practising Jews in 1492. Also, a Spanish creole language known as Chabacano, which developed by the mixing of Spanish and native Tagalog and Cebuano languages during Spain's rule of the country through Mexico from 1565 to 1898, is spoken in the Philippines (by roughly 1 million people).