Lucifer was created by God as a perfect angel. He was called Lucifer while he lived in heaven. After he sinned and persistently refused to repent he was thrown out of heaven. When Lucifer was cast out of heaven, he lost his name Lucifer and he became known as Satan.
In the Dark-Hunter world, at some point—though it is unknown when—Lucifer conceived a son. It is also unknown who is Thorn's mother. Lucifer longs to be free and start the apocalypse, which is why he sends demons to Thorn in an attempt to sway his son into releasing him.