(Greek daimon and daimonion, Lat. daemonium).

The word demon is apparently derived from daio “to divide” or “apportion”, originally meant a divine being; it was occasionally applied to the higher gods and goddesses, but was more generally used to denote spiritual beings of a lower order coming between gods and men. It is now practically restricted to the evil spirits.

A similar change and deterioration of meaning has taken place in the Iranian languages in the case of the word daeva. Etymologically this is identical with the Sanskrit deva, by which it is rendered in Neriosengh’s version of the Avesta. For the original meaning of the word is “shining one”, and it comes from a primitive Aryan root div, which is likewise the source of the Greek Zeus and the Latin deus. But whereas the devas of Indian theology are good and beneficent gods, the daevas of the Avesta are hateful spirits of evil.

Demon is often confused with devil as both qualify the evil spirits or fallen angels. The precise distinction between the two terms in ecclesiastical usage may be found in the decree of the Fourth Lateran Council:

"Diabolus enim et alii daemones" (The devil and the other demons), means that the chief of the demons is called the devil, also found in Matthew 25:41, 
"the Devil and his angels". This distinction is observed in the Vulgate New Testament, where diabolus represents the Greek diabolos and in almost every instance refers to Satan himself, while his subordinate angels are described, in accordance with the Greek, as daemones or daemonia.  
It does not  indicate a difference of nature; for Satan is clearly included among the daemones in James 2:19 and in Luke 11:15-18.


Fra Angelico, Detail of The Last Judgment, San Marco, Florence (1431)

“Devil” comes from diabolus (diabolos — diaballein), the Greek word diabollos which means a slanderer, or accuser, and directly refers to the rebellion of the fallen angel and God’s judgment

"The accuser [ho kategoros] of our brethren is cast forth, who accused them before our God day and night" (Apocalypse 12:10).

It thus answers to the Hebrew name Satan which signifies an adversary, or an accuser.