Sixteenth century Europe was a violent era in history when mass hysteria and religious teachings led people to see witches and demons in everything they did. The fear of hellfire caused self-proclaimed witch hunters to rise up, and among them was the infamous Peter Binsfeld, a man who loved torture and hated Protestants, among many other things. Binsfeld was a respected man in his time, and his teachings on demonology and witches are still read to this day. Perhaps his most well known work is his classifications of the Seven Princes of Hell - seven demon lords who each punished sinners for particular crimes committed during their lifetime.
No one knows what Hell is like, or if it indeed exists, but according to the dogma there is no way to escape from Hell, and the Seven Princes of Hell, as described by Binsfeld, were meant to make people atone for their crimes in life - interesting though they may be, they're certainly not people anyone wants to befriend. Try to avoid it, but if you can't stop sinning you're about to find out who you'll meet in the afterlife.
Lucifer is a name many will recognize. Though the name and its meaning of "morning star" or "light bringer" have a variety of applications in various mythologies, Binsfeld used it to refer to the angel who fell from heaven after his attempt to create a new structure of power. Thus, Lucifer, by Binsfeld's classification, represents the sin of pride. He is often considered to be the leader of demons in Hell, and is therefore particularly evil and dangerous.
Because of his high status and importance, Lucifer is sometimes conflated with Satan; in some stories they are the same, but in others they are separate from one another. Because it's difficult to track down Binsfeld's original work, which was not in English, it's hard to say how he felt about the connection between the two.
Beelzebub is known by a multitude of names, including "Lord of the Flies" and "Baal." The New Testament mentions him as chief of the demons, and Binsfeld associated him with gluttony. Biblically, he's normally associated with diseases - hence the flies - which makes him an odd choice for gluttony.
However, Beelzebub has a long and storied history in biblical lore and apocrypha, where he is often characterized as being just below Satan in rank. Some stories actually rank him as above Satan, but Binsfeld didn't seem particularly interested in hierarchies, just in which sin the demon presided over. His reasoning behind connecting Beelzebub and gluttony are unclear, though it's possible that he's drawing a comparison between flies and the consumption of the dead.
Satan, distinct from Lucifer in Binsfeld's interpretation of Hell's hierarchy, is said to reign over the souls of the wrathful. Because Satan is often conflated with Lucifer, there's some overlap in their histories - Satan is also said to be an angel fallen from Heaven for rebelling against God. 'Satan' is translated to 'the adversary,' in this case the adversary of mankind. Whereas God is loving and wise, Satan is wrathful towards humanity, reflected in his association with wrath according to Binsfeld's categorization.
Often seen as the ruler of Hell, Satan is therefore one of - if not the - most powerful demons.
Perhaps it's not surprising that Binsfeld assigned Belphegor the role of torturing the slothful. His image is one of the least intimidating of the Seven Princes of Hell, as he's frequently depicted on a toilet. However, his past forms have also been associated with phalluses and orgies, but even the Lanterne of Light assigned him to gluttony, not lust. When summoned, Belphegor is said to offer wealth and inventions, and that connection with wealth, which allows humans to not work, is likely why he ended up being associated with sloth. Binsfeld's belief was that Belphegor tempted people to be lazy, and to do evil through their inaction - though he wasn't clear about the means and methods of the torture, it certainly wasn't pleasant.