what do the barrel rating, A,B,C,D, pertain to? I assume it refers to the weight of the barrel-how wide the flats are, but really don’t know for certain.
if so, is A the lightest? Does it have to do with length?
You've got some pretty good responses that address your question, but the answer is a little more straight forward.
The A,B,C, and D ratings pertain to the diameter of the barrel at the breech. In that sense, they relate to the overall barrel size and its weight. That's why the terms "A weight (wt.)", "B wt.", "C wt.", etc. are used.
An "A wt." barrel has a breech diameter of 0.937" or 15/16".
The "B wt." barrel has a breech diameter of 1.000".
The "C wt." barrel has a breech diameter of 1.062" or 1-1/16".
The "D wt." barrel has a breech diameter of 1.125" or 1-1/8".
The profile and length of the barrel will be different depending on the particular barrel pattern, "Lancaster or Dickert", "Golden Age/York", "Jaeger", but the diameter at the breech will be the same for the each letter designation in each pattern.
These barrel profiles have been somewhat standardized among barrel makers so they fit the pre-carved stocks and kits sold by the various suppliers.
This may have started with the Getz brothers as they were the first to develop rifle kits with swamped barrels. They later sold their kit business to Jim Chambers.