The Reaper is an entity that will show up in Mementos if you idle (as in, don't move) on a floor for too long. The Reaper has no weaknesses or resistances, cannot be inflicted by status ailments, and he is a pretty difficult fight if you aren't prepared.
He will typically aim for exploiting weaknesses to get additional actions, and even if you have no weaknesses, he will be sure to Break your resistances. Sadly there's no detailed way to defeat him outside of having a strong level advantage and just overall smart play. I do not recommend fighting him until you are basically at the end of the game.
That being said, there is a very easy exploit you can do to defeat him. It is up to you if you want to resort to this.
Mementos are affected by the Weather, and during Flu Season, enemies have a chance to start the battle with the Despair status ailment. This ailment makes it so the enemy does nothing until their 3rd turn, in which they suicide. Perhaps due to an oversight, the Reaper is not immune to the Flu Season. This means you can start the fight against the Reaper that has the Despair ailment, and he will kill himself (which counts as you defeating him), allowing you to reap the massive EXP reward.
The best way to perform this exploit is enter Mementos on any of the following days: 11/13, 11/14, 11/15, 12/8, 12/9
Go to any floor (preferably one you're overleveled for, so enemies don't aggro to you), stay still and wait the Reaper spawns
Engage the Reaper (do not let him Ambush you)
Check if he has Despair
If he does not have Despair, escape from battle (you'll want to have the Star Confidant's ability to guarantee this)
Repeat steps 3 and 4 (the Reaper will respawn near you after you move)
If he does, simply do whatever until he kills himself
All credit to xNinjaOfChaosx