Castellano is an alternate name for the Spanish language. There are (and were) several romance languages spoken on the Iberian peninsula. Castellano arose in Castilla, Castile, the land of castles (castillos), a large central region of Spain. The alternate name for language is used to contrast with other languages ¿Hablas catalán en casa o castellano? Or, in some Spanish-American countries, it is preferred name for Spanish, especially in academic contexts.
A book of poetry by Blas de Otero: En castellano. Meaning, in plain language. Like "en román paladino," in clear language, according to Berceo: " Quiero fer una prosa en román paladino / en el qual suele el pueblo fablar a su veçino..." It is a bit like asking your doctor to speak in English to you when he or she gives you medical jargon.
Romance meant Latin, but then it came to mean the vernacular derived from Latin. From this we get the word for a novel (in French and German, Román), genres called romances (ballads or prose tales), and romanticism itself.
Usually, when we talk about Spanish in contrast to other languages outside of Iberia, we talk about español.
Castilian, in English, is is the name of a dialect of Spanish, spoken mostly North of Madrid. If you go to article on Castilian language on wikipedia, and switch languages to Spanish, you will find an article on "dialectos castellanos septentrionales."
La Mancha is part of Castile, but the Spanish of La Mancha has some characteristics of the Southern dialects, spoken in Andalusia, Murcia, parts of Extramadura. Madrid is a special case, because nobody is from there. There are a lot of Andalusians but also a lot of Castilians. For that matter, non-Catalan residents of Barcelona are likely to be Andalusians in origin.
Castilla La Vieja was Northern. Castilla La Nueva, Southern. After the constitution of 1978 these became Castilla-León and Castilla-La Mancha, with Madrid, formerly part of New Castile, squeezed between the two. Northern Castile was "old" because of the Southward movement of the reconquest.