The nut is the white block at the upper end of the fretboard that the strings pass over before they get to the tuners.
How wide it is is just that...how wide is it from side to side.
The wider it is, the wider the neck of the guitar is.
Some folks like narrow necked guitars. Some like wider necked guitars.
Narrower necked guitars tend to have the strings closer together. Wider necked...further apart. (The bridge spacing at the other end also determines how wide apart the strings are...and how close together the string slots are cut in the nut.)
No. It doesn't affect tone.
It does affect how comfortable a guitar is to play (for many folks.)
You may want to check out this site.
Full of information about guitars.;
http://www.frets.com/FRETSPages/pagelist.html