A variance is an permitted deviation from an ordinance or regulation. They are generally granted based on a showing that the ordinance or regulation creates a hardship that is unique to your property, in other words not something common to everybody in the neighborhood. They are generally hard to get if the local government considering them follows the law, but that can vary from place to place and some are more flexible than others. An classic example of a variance based on a unique hardship is a variance from a setback requirement, such as where a building is required to be set back from a road or property line by so many feet, but you are the only person in the neighborhood who can't build your house if you complied with the set back because your lot is smaller than everybody else's.
I would think it would be a rare thing to get a variance for chickens. What you need to allow chickens is a change in the law, not an exception to it just for you.
UGCM