Demon Prince

The Demon Prince, Demon from Below and Demon in Pain are the first bosses you'll encounter in the Dark Souls 3 The Ringed City DLC. They occur after the Earthen Peak Ruins and you must kill them to progress with the DLC.

The way this fight works is that you have two bosses to defeat at first: The Demon from Below and the Demon in Pain. They both share the exact same moveset, but one starts off in a fire form and the other starts off in a poison form.

It is crucial that you defeat the Demon in Pain (the one not on fire when you start the fight) first because if the Demon from Below is killed first, the entire fight will be significantly harder. Once both Demons are dead, the last Demon will transform into the Demon Prince, which will be very similar to the other Demons but have a slightly altered moveset as well as some new moves. 

If the Demon from Below is killed first, the Demon Prince will have many fireball attacks including a giant meteor attack that will most likely hit you and kill you. It is incredibly difficult to dodge, almost impossible even at the time of writing, and is the reason you shouldn't kill Demon from Below first.

If the Demon in Pain is killed first, then the Demon Prince will be almost exactly like the Demons from before except he will have a giant laser attack that is incredibly easy to ignore if you just get behind him and it will also give you a good chance to attack him during this moment.