That British map is historical and very much out of date though - places such as Huntingdonshire and Westmorland are long gone, Sussex hasn't been one county for about 160 years, and Scotland and Wales have had complete overhauls.
For an up-to-date list, check out the right-hand column of the table on this page:
http://www.peak.org/~jeremy/dictionary/tables/ counties.php
So that's England 47 counties, Scotland 32 and Wales 22. Together with the 6 in Northern Ireland, that brings the total to 107.