The following list is based on a combination of in-game sources and fan conjecture - many enemy families have not been named or classified by official Square Enix material; the current classifications are merely placeholders.
Standard = Enemy will spawn naturally without any prompt.
Boss = Powerful enemy that appears as a boss in the associated dungeon, trial, or raid.
Boss minion = Enemy will only spawn as part of a boss encounter, and will despawn upon the defeat of said boss.
Quest, levequest, FATE, guildhest, treasure map = Enemy will only spawn when prompted by the associated event.
Quest, levequest, guildhest or FATE boss = Powerful enemy that is the sole or primary target of a FATE, levequest, guildhest, or instanced quest.
Levequest wanted target = Powerful bonus enemy that occasionally spawns during associated levequests that is not part of the levequest's objectives, but drops valuable items.
Elite mark = Powerful enemy that will spawn as part of the Hunt, with its strength and spawn times and conditions varying based on its rank.
Enemies found in the Diadem hold ranks based on their difficulty, the ranks being I through V with an additional ★-rank signifying the toughest enemies.
50-60*Odin's level in the "Steel Reign" FATE varies between 50 and 60. If the FATE is failed, Odin's level will drop by one the next time it spawns, and vice versa if the FATE is completed successfully.