A nut size is determined by the size of the threaded hole. Thus, a
1/4-20 nut has a nominal 1/4" hole with 20 threads per inch. Metric
nuts, same story, but pitch/size is expressed differently.
The wrench and/or socket you use bears no clear relationship,
obviously. The size of the wrench you use is whatever fits snugly, and
is expressed as the size across the flats.
I think the Witworth (semi-obsolete UK system) measures across the
points. I remember on an old Austin-Healy (gee, how I miss that car!)
that I had problems with my US wrenches fitting properly.
For US nuts/bolts, use fractional inch wrenches and/or sockets. For
metric ones, use Metric wrenches/sockets. Some sizes are pretty close.
I notice that the last set of 3/8" drive sockets I picked up had both
fractional inch and metric sockets in the set.
You need both, as I've found that my 2000 Jeep Cherokee has both
metric and US fittings!