So basically, Hazbin Hotel takes place in Hell and the main plot is that sinners go though the Hazbin Hotel (formally Happy Hotel) to redeem themselves and go to Heavens. Why? Because of the yearly exterminations by Exorcists to lower overpopulation in Hell. The hotel was thought of by Charlie, Princess of Hell and with the help of her Girlfriend, Vaggie they had their first client, Angel Dust, a porn star who agreed to stay at the hotel rent free if he didn't do drugs, cause trouble, or do anything related to sin. But after their broadcast at the 666 news station went very wrong after Charlie discovered Angel broke his 2 week clean period by helping his friend, Cherri Bomb with some turf war after the latest extermination. After the train wreck happened, they're back at the hotel, Alastor comes to visit the hotel because he's bored and after he takes a look around and given permission to leave the hotel whenever he pleased (because he wanted to see the sinners who wanted to be redeemed, only to fail their attempt), he summons 2 helpers, Nifty, a small demon cyclops who died in the 50s and Husk, a winged cat demon who drinks a lot and gambles a lot. After he (Alastor) sings a bit, Sir Pentious comes back after the turf war against Cherri Bomb and Angel Dust, by blowing a hole into the hotel and his airship is destroyed by Alastor's shadow manipulation and the cast goes back into the hotel and the Happy Hotel is renamed, the Hazbin Hotel.
Also, there's a sideshow named Helluva Boss that takes place in the Hazbin Hotel universe, you can expect it to be just as 18+ as the main show.