Bestiary

The Bestiary is an in-game, filterable encyclopedia that shows information about enemies, critters, bosses, and NPCs. It can be found by opening your inventory and clicking the book icon located to the right of your item trash can. It includes basic statistics like health, damage, defense, and knockback, as well as advanced information like item drop rates. Furthermore, it displays a short description of the entity similar to item tooltips. Entries can be sorted by different statistics and can be filtered by certain locations, times, events, and other categories.

Initially, no information is available about any entity, and there are four stages of progression: Progression is done through three different counters, depending on the NPC's nature: enemies have a kill counter (which is separate from the one provided by the Tally Counter, so kills made before the 1.4.0.1 update do not count towards filling the bestiary), critters have a proximity counter, and Town NPCs have a chat counter.
 * 1) Name and portrait only (1 Killed)
 * 2) Add stats (15? Killed)
 * 3) Add drops (25 killed)
 * 4) Add drop rates (50 Killed)

For common enemies, a total of 50 kills is needed to complete information. Note that enemy with different variants (such as zombies) have separate trackers for each variant (50 arrow zombie kills, 50 gel zombie kills, etc...). This does not apply to bosses, as killing a boss once unlocks all information on it. At the bottom of the Bestiary is a bar that fills as the Bestiary is completed, allowing players to track their completion progress. Bestiary progress is tied to a world, not a character.

Filling in part of your Bestiary allows you to unlock the Zoologist NPC, which sells more items as you continue filling in the Bestiary, such as the Cat License. Unlocking all the information for the entries is not a requirement for this, as long as you unlock the entries themselves.

Three monsters are mutually-exclusive: the Crawdad, Giant Shelly, and Salamander. In a given world, there can be only two of them, chosen randomly; the third will not appear. This is taken into account by the bestiary, and the missing monster's unlock status will be equal to the lowest unlock status of the rest of the bestiary, allowing to completely fill it anyway.

Tips

 * You need to fill out 10% of your Bestiary (52 entries) to have the Zoologist spawn.
 * The Zoologist's items sold depends greatly upon the percentage of Bestiary completion. This includes items like the Leather Whip, Jousting Lance, and Universal Pylon.
 * The Bestiary is saved on a per world basis.
 * Once you talk to an NPC, you can find them in the Bestiary, which sheds some light on their preferences. In case you have not defeated the Dungeon Guardian before defeating Skeletron, you can get his bestiary entry by talking with the Clothier.

Trivia

 * The first person to complete the Bestiary was a YouTuber named Hectique. He has uploaded a full video showcase of the bestiary here.