Apollyon is actually the Greek name for Abaddon, and it means "The Destroyer". In early Christian literature it was a name for the devil. He is identified as an angel of death, "hideous to behold, with scales like a fish, wings like a dragon, bear's feet, and a lion's mouth."
Apollyon is aligned with the Deva Cult and is found in Ikebukuro, where it is guarding Hiroko. It drops an Inverted Cross upon its defeat.
Devil Summoner 2: Raidou Kuzunoha vs. King Abaddon
Originally called Destroyer Insects by Dahn, Apollyon appears as a horde of invincible Locust demons from the Infinite Abyss that obey King Abaddon. Originally only one was found in the capital but thanks to Dahn's actions the Pojitrawn became complete and a portal leading to the Infinite Abyss opened up in the sky turning it red as aresult. This unleashed the swarms of the Apollyon horde upon the capital. Dahn would attempt to control them only for them to fail to listen to him.
Eventually stronger variants called Psycho Apollyon (サイコアポリオン?) begin to appear after humans start giving into despair. They are the manifestation of mankind's negative emotions once they've given into their despair. To add to the threat the Apollyons represent, they are capable of simply regenerating any damage dealt to them faster than Raidou can hit them, rendering them virtually invincible. This can only be countered by creating the poisonous Youmetsutou sword, which can negate their healing ability with its Apollyon Killer power.
Talking to certain demons reveals they are also in danger thanks to the Apollyons because they are no longer able to get the energy they need to survive. Apollyons will always target those who carry Luck Locusts on them. The most reasonable conclusion for why is that since luck and Apollyons are opposites they attempt to get rid of that which is a threat to them.