Demon Horns are items that are obtained by defeating normal NPC demons (e.g no blue demon, no boss demon..etc). They aren't a guaranteed drop, but there's a high chance that the demon killed will drop one. Defeating a higher NPC demon (demon with a Blood Art) will guarantee a Demon Horn drop 100% drop rate. Currently, only the Flame and Wind trainers need 20 Demon Horns for their breath quests. In the Final Selection, you will most likely obtain more than 20 Demon Horns, as you have to kill 30 demons to complete the exam. Demon Horns can also be sold for a decent amount of Yen, 25 Yen per horn, at the Slayer Corps HQ. Demon Horns are droppable. They give no benefit to Demons, other than trading.
Note:demons can sell demon horns as long as they find a way in Slayer Corps.
A picture of the Demon Horn Item in your inventory.