Mimicology Quests are Sidequests assigned by Cypress that involve collecting a certain number of specimens (e.g. body parts) which are dropped from Mimics, left over from the burn-up process. Receiving a specimen is not certain when most mimics die so you may get other items; the exceptions are the special studies which add mimics on the ground.
The first Mimicology Quest becomes available after your character kills a Firegrub. After delivering the specimens, Cypress will need one day to study them before the quest can be completed. The "day" is a Nidarian day; you can deliver the specimens at 23:59 NT and complete the quest at 00:01 NT. The reward for most of these quests is 30 XP each and a Sprocket with information on the related Mimic.
After the first, studies are chained so each requires a previous one to be completed. Each also requires having killed at least one Mimic of that species to access the quest.
The yellow rows are a separate chain for Mimics only found on NG-only Co-op Trails. Non-NG players can receive these studies after killing one during a Co-op on the member-only Trails. The red rows are Mimics only found in NG Daily Quests and cannot be completed by non-NG players.