From Diablo Wiki
A Fetch Quest involves, more or less, fetching something for an NPC quest-giver. The item being fetched is irrelevant, as the player will simply be fetching something. This type of quest is also referred to as a "FedEx" quest.
Get it, boy!
An example of a fetch quest in Diablo III would be an Act I quest entitled Little Girl Lost, which involved fetching a ghastly doll for a ghostly little girl. That pretty well sums up a fetch quest in a tidy fashion. Often, a fetch quest and a kill quest can overlap. The way to tell if one is a fetch or kill quest is basically decided on if an item needs to be taken from the corpse. If so, it is a fetch quest, even though the fetching of the item involves killing. If not, it is a kill quest. Sometimes a fetch quest will require the player to fetch multiple things, such as in Diablo II with the Khalim's Will, or the Horadric Staff in Act II of Diablo II.
The fetch 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 if the game is an MMO or a single player game. The other two types of quests are called escort or kill quests.