A county was originally a political and administrative division within a state. This is still its main meaning. The word derives from the Anglo-French "counté", which in continental European usage meant the area controlled by a Count.
United Kingdom usage
Due to the reorganisation of local government there are now several different uses of the term. There are still administrative counties, in which the County Council has responsibility for some, or in some cases (e.g. Wiltshire), all of the functions of local government, apart from the very minor powers exercised by parish councils. Those which are not unitary authorities have district councils (sometimes called boroughs within them.
The "geographical counties", while having the same name, may extend over a wider area. For instance, the geographical county of Devon includes Plymouth and Torbay which do not come under the county council.
The "historic counties" are different again. For instance, the present county of Somerset formerly included parts of the relatively new county of Avon while Kent and Surrey included parts of Greater London, while the short-lived county of South Yorkshire, which is still a "ceremonial county" with a Lord Lieutenant, was formerly part of the West Riding of Yorkshire.
In Scotland administrative functions are no longer exercised at county level, the regions having been superimposed.
Yorkshire, a very large county, was formerly divided into three "ridings", East, North and West for administrative purposes.
The term "county borough" used to apply to boroughs which did not come under a county council. These are now normally referred to as unitary authorities.
Although metropolitan counties do not normally have any real administrative functions (these being exercised by the metropolitan boroughs), the exception to this is Greater London which has a Mayor with strategic responsibilities, and a Greater London Assembly, intended to hold the Mayor accountable. However, Greater London, like the Scilly Isles, is not officially a county, being instead sui generis.