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.
Spanish language
Spanish is by far the most common second language in the US, with over 50 million total speakers if non-native or second-language speakers are included. While English is the de facto national language of the country, Spanish is often used in public services and notices at the federal and state levels. Spanish is also used in administration in the state of New Mexico. The language also has a strong influence in major metropolitan areas such as those of Los Angeles, Miami, San Antonio, New York, San Francisco, Dallas, and Phoenix; as well as more recently, Chicago, Las Vegas, Boston, Denver, Houston, Indianapolis, Philadelphia, Cleveland, Salt Lake City, Atlanta, Nashville, Orlando, Tampa, Raleigh and Baltimore-Washington, D.C. due to 20th- and 21st-century immigration.