Quest are different in-game actions with rewards. There are quests for construction and improving buildings, attacking pirates and other players, researching technologies, creating fleets. You will receive accelerators, ships and resources as a reward.

With the 1.20 update, the new types of quests are available:
-- Daily missions: restart everyday;
-- Weekly missions: restart once the player gets the reward;
-- Monthly missions: restart once the player gets the reward;
-- The starting missions: do not restart once a player gets them done and receives the reward.