A shire is an area with its own government and police force, usually part of a larger region or country.
You might hear about places like Hampshire Shire in England, which is a big county with its own local authorities. In the UK, these areas are often called "shires" because they were originally self-governing regions ruled by a lord or earl. Today, shires usually have their own councils that make decisions for the people living there, like setting taxes and keeping public spaces tidy. Some countries use similar terms to divide up their territories, so you might hear about states or provinces in other parts of the world being compared to shires in this way.
An administrative area or district between about the 5th to the 11th century, subdivided into hundreds or wapentakes and jointly governed by an ealdorman and a sheriff; also, a present-day area corresponding to such a historical district; a county; especially (England), a county having a name ending in -shire.
Example: Yorkshire is the largest shire in England.

And then finally, is it true that you let out the fart heard round the shire, doing a hillside roll with your Hobbit castmates?
8:02And there was also the cart. So when I jump into the cart at the very beginning of the film, when Gandalf's coming into the Shire for the first time and, you know, you're late and the wizard's never late.
12:45