Favors are tasks that villagers might ask the player to do. They may ask the player for shirts, fossils, or bugs. Depending on their personality they can also ask the player to deliver gifts and letters.
An option will appear when talking to a villager, that says something along the lines of "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/her later to get one from that particular villager. Villagers will usually ask the player to get a Pikachu doll or Gameboy from one of the villagers. The villager in question may have let another villager borrow it, making things more difficult. Other jobs might be giving clothes or furniture to another villager, finding a ball, or something else.
In Wild World and City Folk
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 this favor cannot be done. If the player opens the present or read the letter before it is delivered to the villager, the villager might ask why it was not wrapped. Most of the time these favors will have a time limit, some of them short, such as ten minutes, or some a long stretch of time, such as before midnight.
Other times, villagers will ask for things like clothing and fossils. Snooty villagers may ask the player to bring them furniture or fish, because it is in style. Saharah, the carpet saleswoman, may ask the player to deliver carpets to Tom Nook, Pelly, Phyllis, Blathers, Mabel, or a random villager.
In New Leaf
Similarly to past games of the Animal Crossing series, villagers will ask the player to deliver a present, letter, bury a time capsule, or return an item to a friend for them. Depending on the favor, if the player's pockets or message box are full, they will not be able to do the task. Also, the same rules apply about opening the gifts or reading the letter: villagers consider it to be rude, and the receiving villager will ask/complain about the condition of the item. The most common interval is either within the hour or "by the end of the day." Other favors are more personal; new clothes, fossils, fruit, and furniture. Completing each task for a villager results in an improvement in the player's relationship with that villager. Villagers may also give presents to players who return to them after completing the task.