Tainted monsters are special optional boss-like monsters with a purple darkness swirling around them. You can find them on the field and in dungeons.
The last of the 50 tainted monsters is level 69 so I'd recommend being around level 60 or so. There isn't much strategy to beating them apart from adjusting your tactic tweaker to their monster type and affinity and being an appropriate level.
Some tainted monsters only appear during quests so if it's not showing up in the tainted monster menu then it's part of a quest you haven't gotten yet.