From Diablo Wiki
A Kill Quest is, quite simply, a quest that involves killing something. A kill quest can involve killing one specific target, or it can involve killing a certain number of targets.
Killing as a Business Opportunity
An example of a kill quest in Diablo III would be the Act I quest entitled The Skeleton King. The objective of this quest is to kill an individual named Leoric that, in all likelihood, probably has it coming. Again. In many games, a quest to kill a specific person or monster is often given to the player through a system such as a Wanted Poster, or sometimes it comes from a sort of "bounty board" within an NPC town or village. Sometimes the player must return to the quest-giver, or someone responsible for the bounty reward in order to finish the quest. In other instances, proof of death is required, coming in the form of various body parts taken from the slain opponent. If the quest requires the player to bring back an item, then it is considered a Fetch quest.
The kill quest is one of three types of quests that makes up what can be called the "Quest Trifecta" in RPGs. It is called such, because there are generally only three types of quests in an RPG, regardless of the game is an MMO or a single player game. The other two types of quests are called fetch or escort quests.
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