Legendary hunts are basically mini boss fights found in different locations around the the different areas. There are 10 battles to complete in total (please see below for more details) and you can check to see if you have already completed the hunt for that area as the legendary hunt will be a challenge you can check in the map.
I would highly recommend going for this achievement once you have completed the story so all of the areas are unlocked and the hunts are easy to take down.
Pandora:The Droughts 0:00
Ascension Bluff 0:40
Devil’s Razor 0:57
The Splinterlands 1:35
Konrad’s Hold 2:56
Athenas 3:32
Eden-6: Floodmoor Basin 3:59
Voracious Canopy 4:30
Nekrotafeyo: Desolation’s Edge 4:50
The Pyre of Stars 5:17