The monster hunt quests are given by Reorak the Tall in the Hall of Champions in Lormidia. The Monster Hunt Quests are generic quests - like the Caravan Escort quests. You can keep on doing them but just one at a time.
There are four difficulty levels in the monster hunts. You need to reach a certain level before you can get them. To complete a monster hunt you simply need to kill a certain number of a certain monster type. The monster type depends on the diffculty level of the hunt. See below.
By doing Monster Hunt quests you can gain trophies. As soon as you get your first trophy you can get the Champion's Vigor blessing at the Hall of Champions.
Notes: level requirements seem to be removed now. The lists below are incomplete.
Easy Monster Hunt
Minimum level: 10
Minimum level: 15
Difficult Monster Hunt
Minimum level: 20
Extreme Monster Hunt
Minimum level: 25