Quests are missions or tasks that the player is given throughout each installment of the Diablo series. Quests usually serve to advance the game's plot or expand upon the elaborate storyline. Most quests give rewards for completion, be it an item, statistic enhancement or other benefit.
In Diablo I, the quests are a series of missions (fewer in multi-player) on which the character is sent, and which offer advantages when completed. None of them need to be completed, but all add background detail and most give valuable rewards (even if only when reselling). Though there is only a finite number of unique quests, each new game generates a random selection of quests. Therefore if players create a new game they might encounter different quests.
In Diablo II, the quests are a series of missions of which most are integral to proceeding in the single-player game. For example, without the Cube, the Heart, Eye and Brain, and the Flail, the single-player character cannot get beyond the Third Act City. However, in multi-player, an advanced character can open a town portal from a waypoint beyond the blocked part, enabling a less-experienced player to bypass the quest entirely. Each Act has 6 quests, with the exception of Act IV, which has only three.
Diablo III promises to bring more interactive quests that will be much more rewarding. They tie in closely to the idea of randomness - when replaying the game, the player will most likely get to find quests that were not available to them before.