The Sovereign of Hell is the absolute overseer/ruler of the realm known as Hell, which serves as the eternal home of demons and sinful deceased mortals. The sovereign is referred to as the Lord of Hell, King of Hell, and King of Hellfire.
The ruler must be of God's lineage (an angel) to prevent chaos. Nonetheless a demon can serve as second in command, a notable example being Mazikeen. Charlie, as a half-human/half-angel, could serve as king, as it shown that Dromos planned to mold Charlie into becoming the king he wanted. Aurora, a nephilim like Charlie, and Lucifer's own daughter, could serve as ruler as well.
God later explains Hell no longer needs a warden, albeit to Amenadiel off-screen.
Lucifer later retakes his role in a new capacity to help heal the guilt of the damned. Chloe becomes consort to the sovereign of hell upon her death.
The King of Hell oversees the realm. He is responsible for controlling demons, ensuring damned souls get punished, and keeping the kingdom contained. He must also have his servants (demons) patrol the gates of Hell, making sure damned souls do not escape.
The first ruler of Hell was the angel Lucifer (formerly Samael), who was cast out of Heaven as punishment for rebelling against his father, God. Lucifer unwillingly ruled over Hell for billions of years, even long before there were damned souls, before deciding to abdicate his throne and leave Hell for Earth.
In Lucifer's absence, Amenadiel was forced to run Hell, a job he loathed. Amenadiel was unable to rule Hell because the demons would not take orders from anyone but their king Lucifer. Amenadiel tried numerous times to convince Lucifer to go back to Hell but was unsuccessful.
Lucifer had suggested to his mother, the Goddess of All Creation, to become the new ruler of Hell as opposed to her original plan to return to Heaven.
When Michael began impersonating Lucifer on Earth Amenadiel went to hell to inform Lucifer who then went to confront Michael while having Amenadiel temporarily fill in for him as ruler of hell until God relieved him of that duty and claimed that hell no longer needs a warden to keep it in check, making the position of Sovereign of Hell either optional to a celestial who would want the position or perhaps the position itself has been completely abolished.
At the end of season 6 the series finale, Lucifer realized what God meant by Hell no longer needing a ruler when Lucifer realized his purpose in being Hell's healer instead of jailor.