Quest fish are the special catches the Angler asks for — one new target at random each day. They only bite while their quest is active and are turned in for coins, gear and vanity rewards rather than cooked or crafted.
There are 41 Angler quest fish. The Angler asks for one at random each day, and catching and turning it in counts toward his reward milestones.
Each quest fish spawns only in a specific biome and depth layer while its quest is active — the Angler's dialogue hints at where. Fish that spot with a pole and bait until the named fish bites.