From Nookipedia, the Animal Crossing wiki
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.