If you have played previous entries in the Persona series, then you will know that the dungeon areas of the game have a variety of rare monsters hidden in their depths. In Persona 5, these rare monsters are known as Treasure Demons, and can be encountered by attempting to steal the many smaller treasures hidden around each Palace. Once they appear in the form of a large red gem, the player must attack the jewellery to initiate the battle with the elusive monsters.
Defeating a Treasure Demon, or adding it to your party by taking it’s mask has quite a few benefits. You will gain a large amount of experience points for the defeat, usually averaging around 1,500 EXP, but this can be increased if you have built up your Moon Confidant by interacting with Yuuki Mishima. A Treasure Demon cannot be used in battle, but it can fused via other Personas, and some of them have highly desirable support skills which can be inherited by the resulting Persona if you choose to allow it to happen.
If you happen to let a Treasure Demon escape, do not worry. Once you have captured, defeated or encountered a specific breed, it will re-appear in Mementos as a random encounter. When you encounter them in this way, the specific type of Treasure Demon will be chosen at random, so you might have to search for a long time if you require it for a fusion recipe later in the game.
In case you want to go hunting for them, here is a list of all the Treasure Demons you can encounter in Persona 5, as well as the most effective methods of defeating them in battle.