Quests are missions that random ships are chosen to accomplish. These ships will leave the galaxy until their quest is done. They will return with a gift as a quest reward if you choose to welcome them.
Gift of Exhibition
Spawns neutral Explosive Museums on the right side of the galaxy.
Gift of Eyes
Spawns a Shoggoth.
Gift of Friendship
Spawns an army of neutral Raiders on the left side of the galaxy.
Gift of Ignorance
Sets tech level of all empires in the galaxy to 1.
Gift of Guns
Spawns a group of common ships in the empire that sent the ship on the quest.
Gift of Warmth
Spawns a Dragon.
Gift of Chase
Spawns a group of Hunter in the middle of the galaxy.
Gift of Arms
Spawns an Elder Thing.
Gift of Knowledge
Either sets empire of owned ship to tech level 8 OR raise tech by one. The latter effect is much more common than the former, especially if transcedence is disabled and empires are allowed to go beyond tech level 9.
Gift of Comradery
Removes all peace.
Gift of Perspective
Turns your view of the galaxy 360 degrees.
Gift of Madness
Sets every empire's state to ALL_WILL_BE_ASHES. This effectively destroys all life in the galaxy. Very useful if you want to reset your galaxy.
Gift of Vision
Makes everything look creepy for a while.
Gift of Mother
Gives a mothership to the the empire that owns the ship on the quest.
Gift of Calm
Spawns the Quiescer. Very useful if you want to reset your galaxy.
Gift of Nature
Spawns the Yggdrasil
Tenfold Gift
Spawns a Cecealian Abomination, which will destroy stars while growing tentacles.