Maps of UK Administrative Areas

Click County Map or District Map to get a list of unitary authorities, and either counties or districts.
This will give a map, and list of wards, and parishes if there are any.
Click through to a parish or ward map

There is a lot of historical complexity!

Some parts of England have 2 tier local government (County and District).
The rest of England, Scotland, Wales, and Northern Ireland have a single tier Unitary authority. e.g. Cornwall is a unitary authority, but neighbouring Devon is 2 tier, with a county council, over several District Councils.
Many English unitary authorities are cities, split off as county level unitary organisations from their surrounding county.
London has a 2 tier structure - the Greater London Authority which is sub-divided into Boroughs
For electoral (not administrative) purposes, both Districts and Unitary Authorities are split into Wards, which elect local councillors.
Separate from Wards are Parishes (though their areas often the same). These are not church parishes, but administrative areas with a Parish Council, which has a local voice on Planning matters.
Finally, what Brits normally think of as geographic counties (e.g. when addressing a letter) no longer really exist for administrative purposes. In 1972, the historic counties were rearranged, and split into urban and county areas [wikipedia]. An example is Middlesex, which no longer exists. The historic counties map shows them.
Not shown, are "ceremonial counties", which are also based on the historic county structure, which areas with a Lord Lieutenant.
Wikipedia explains this bizarre complexity

Developers

The boundary data comes from OS Open Data in very high (1m) resolution ESRI Shapefile format.
A better (much more useful) source is the Office of National Statistics 'data portal, which has lower resolution versions (i.e. much smaller files) of the same data.
I've smoothed the boundaries (i.e. ran a point reduction algorithm on them) to 10-50m resolution, and converted them to geojson/KML