A shire is an area in England that originally was under the control of an Earl, an earldom, e.g. Wiltshire, Yorkshire. They often cover all or part of a county. Both shires and counties were used as formal demarkations within the kingdom's administration, with their own reeves and sheriffs.
Strictly speaking, it is a tautology to say the County of Yorkshire (not least because, in period it was divided into thirds (called ridings from the Norse for third-part)), but it is occasionally done. Really it should be Yorkshire and the County of York.
Shires in the SCA
A shire is a local branch of the SCA which reports directly to a Kingdom or Principality. The minimum requirement for a group to become a Shire is 5 sustaining-level financial members, and a petition that shows that the majority of the members of that group want the elevation.
In the early days of the SCA, and in areas where population density is low, shires often spring up out of nowhere. As the amount of unclaimed (for SCA purposes) land goes down, one finds shires splitting off from a Barony or a Province, often after having been a Canton or a Riding first.