A petition form is an item given to a player by either Dr. Shrunk or a villager in Animal Crossing: New Leaf with which the player needs to obtain 6 signatures from villagers in order to complete the request.
Occasionally a villager will ask the player to get 6 signatures from villagers in other towns to gain favor for an idea they have. If the player takes too long to fulfill the request, the villager says that they had to mail the list off already and so it was too late. This makes the petition form turn into a Paper Scrap, a trash item.
The petition form is an item that cannot be dropped on the ground, put into storage, thrown to the trash can, or sold.
You can check the petition form at any point to see how many signatures you have and how many are remaining.
If the player visits another town with a petition form during that town's Fishing Tourney, they will be unable to talk to the villagers about the petition or get signatures.