Quests are small tasks given by RNG and, for one time, the minion. Completing these will grant you XP, although they are otherwise pointless and amusing, as even RNG admits. Still, they provide a little extra challenge that can add some fun to a level. Quests can be viewed in the Lore tab on the menu.
RNG only ever gives two types of quests. One of these involves gathering a randomly-generated amount of gems from around the level. These are red and can be picked up like normal items, although they will not be visible in your inventory. The only reward for completing this is the quest XP.
The second type of quest is to kill a monster. This will summon a new monster somewhere on the level, which will be highlighted. This will at least give you another monster on the level and some additional XP from that, as well as quest XP.
For the quest given by the minion, see the relevant section of the minion page.