This mechanic is fairly straightforward and is what gives purpose to the game. The quests give players a summary of what they have to accomplish in order to receive a reward. This is done in nearly all games and sometimes can even work alongside tutorials. For example, the player must perform a certain number of actions until they have completed that task and can then move on.
A good example of a game that uses this mechanic is the MMORPG Elder Scrolls Online. The player receives quests by talking to npcs. The quests then cause players to use other mechanics such as the discovery as they have to venture out into the world to complete these quests. This then works alongside the reward mechanic as when they complete the quest the receive a reward. This could be in the form of an item or even experience that levels up the player.
Another game that uses this mechanic is Grand Theft Auto. The player makes their way through a story which is triggered when the enter a mission start zone. After the cut scenes have ended the player is given information about where they must go and what they must do. This is a form of quest. When the player completes what is required the mission is completed and the player receives money which they can use to buy new weapons and vehicles.
If I were to use this mechanic it would also be in an MMORPG as it is a good way to give the player direction on what area of the map they should be in depending on their level. It also allows them to gain better equipment for these new areas and experience to level up.