Favors are requests from villagers. They might ask for a shirt, a fossil (a specific fossil or just one in general), a bug, a fish, or a piece of furniture. Depending on their personality, they can also ask the player to deliver presents and letters.
In Animal Crossing
When the player speaks to a villager, they have the ability to ask to perform a favor. The option will be something such as, "Need help?" or "Can I have a job?" The villager will often have something for the player to do, but may also have nothing, which means the player may have to talk to him or her later to get one from that particular villager. Villagers will usually ask the player to get things for them from one of the villagers or to deliver something for them. The villager in question may have let another villager borrow it, causing the player to have to obtain the item from whoever it was given to.
Sometimes a villager will ask the player to deliver a present or a letter to another villager, but if the player's pockets or letter slots are full, the favor cannot be done. If the present is opened or the letter is read before it is delivered to the villager, they may be ask why it wasn't wrapped. After the delivery is done, the player will get a reward, such as Bells or furniture. Furthermore, in Wild World, when the player taps the villager who gave them the favor, they will frown rather than wave. Most of the time these favors will have a time limit, some as short as 10 minutes and some a long period of time (may be in the form of a time limit, such as "before midnight"). Other times villagers will ask for things like clothing, fish, bugs, fossils, and furniture that is a specific color or from a certain series. In New Leaf, the villager will instead simply ask for furniture, fish or bugs. The time limit is also longer, meaning that the player has to do this before the villager goes to bed or, in the event that a villager is in their house and has the player get another villager to their house, within one hour. If the time for a favor for a fish, bug or fruit expires and the player brings the requested item, the villager will say that they forgot about the favor and either caught a fish or bug of their own or they got a requested fruit themselves and will apologize, telling the player to do it faster next time. The letter delivery was also removed in New Leaf, being replaced with a request to bury a time capsule that must not be opened before it is buried or the player will fail the favor. The failure will be more negative if the player takes the item included in the time capsule out of the capsule after reading it.