First: a lot of demonology found in the grimoire tradition comes out of Christianity. I’ll do grimoire-related demons later on, so let’s just pretend they don’t exist for now, okay?
There appear to be two kinds of demons in Christianity (ymmv; not all forms of Christianity do this). The first is the unclean spirit and the second the fallen angel.
There are about 20 mentions of “unclean spirits” (pneumata akatharta) in the New Testament, such as the one’s Jesus casts out into swine that then kill themselves. There are also the “air spirits” (aero tou pneumato) of Ephesians 2:2 Presumably those spirits are demons of some sort, but of what variety it’s hard to say; the term “daimon,” the source of the English word “demon,” isn’t used, nor are these described as fallen angels. The movement, Christianity, of the morally neutral term daimon into something evil has already been discussed in the etymology part of the Guide. It is perhaps noteworthy that daimon only shows up only about five times in the New Testament, as opposed to the 20 or so times pneumata akatharta appears.
It is interesting that there is no direct connection, Biblically speaking, between these spirits and fallen angels, fallen angels that show up in . . . a bunch of different places, actually. For instance there is 2 Peter 2:4 “For if God did not spare the angels when they sinned, but cast them into hell and committed them to chains of deepest darkness to be kept until the judgment” and Jude 6: “And the angels who did not keep their own position, but left their proper dwelling, he has kept in eternal chains in deepest darkness for the judgment of the great day.”
The Apocalypse of St. John (aka Revelation) is another source for fallen angels. Importantly, the idea of the “War in heaven” comes from here, though it was likely heavily influenced by the apocryphal 3 Enoch, which describe a similar war between the sons of light and the sons of darkness. This will also be connected to the vision of a star falling from heaven, to be discussed a little bit later. The war is first discussed in Revelation 12:7-9, where the angle Michael boots Dragon, identified with Satan, from heaven, along with Satan’s angels.
An earlier portion of Revelation, 9:1, will eventually be connected with the figure of Satan. Rev 9:1 describes a star fallen from heaven to the earth (just as Satan is cast to the earth by Michael in Rev 12) and who has the keys to the abyss (i.e. hell). Due to what is, in my opinion, a misreading of Isaiah 14:12, this fallen star will be given the name Lucifer, to be connected with Satan.
Lucifer is a somewhat difficult but important subject, especially as he is connected to Satan. It seems clear that the part of the Hebrew Bible (Isaiah 14:12) where Christianity gets Lucifer from isn’t talking about an angel, fallen or otherwise. Rather the text is talking about the king of Babylon, who is given the title Day or Morning Star and is described as eventually falling from heaven (yeah, it’s a metaphor). But eventually this will get translated into Greek and from the Greek, Latin, where in some readings the title is turned into a proper name, something that is retained in the King James Bible today.
Finally, at least a few demons are described. Abaddon, for instance. Originally, Abaddon meant “place of destruction” and was associated with Sheol (an afterlife realm) in Jewish writings. In Revelation Abaddon becomes a king of hell and may even be associated with (or as) the Anti-Christ. Revelation also mentions “demonic spirits” that go out and perform miracles. Again, just what demons are is questionable. Another is the Dragon, identified with Satan, with seven heads and ten horns and seven crowns and who probably represents the Roman Empire.