A bestiary is a compendium of monsters. Most games in the Neptunia series contain a bestiary that registers and lists enemies the player encounters along the game.
The Bestiary is a list of monsters that if filled up as the player encounters and defeats them along the course of the game. Bestiaries often include information such as where the monster can be found and what items it can drop. In the case of item drops it is required for you to have actually obtained the item for it to be visible in the enemy's data in the Bestiary. Though you will often be shown how many items an enemy will drop so it's helpful in seeing if you have obtained all the item drops from a specific enemy.