The Path of the Hunter missions in Far Cry 3 are a set of special side-quests associated with Outposts, which require you to hunt down a specific animal or set of animals with a given weapon. These missions are unlocked when the respective Outpost is liberated.
These missions are designated on the map by blue deer-like icons.
Thankfully, the names of the Outposts (including the non-liberated ones) are given on the map, which makes it easier to spot which Outpost will ask you for what kind of Path of the Hunter mission, provided you are aware of the animals.
There are certain Path of the Hunter missions that require you to hunt down very specific unique creatures. These tend to be more demanding (but with better rewards) as compared to the common animal ones.