Pasted description below from Sheldon's page. The A/B/C designation is even more confusing when you consider 700C wheels/tires which these days range from the uncommon 19mm to well over 40mm.
In the French sizing system, tires are designated by a three-digit number, which may be followed by a letter. The number is the nominal outside diameter of the tire the rim was originally designed for. The absence of a letter indicated a narrow tire; "A", "B" and "C" indicated increasingly wider tires."A" was originally a tire about 30 mm wide, so the 650A rim is pretty large, 590 mm. If you add the top and bottom 30 mm tire thickness to 590, you wind up with the 650 mm tire diameter.
The 650C size was originally intended for a quite wide tire, about 40 mm wide. Top and bottom 40 mm tire plus the 571 mm rim size again bring you to a 650 mm outside diameter, even though the rim was smaller.
In the French sizing system, tires are designated by a three-digit number, which may be followed by a letter. The number is the nominal outside diameter of the tire the rim was originally designed for. The absence of a letter indicated a narrow tire; "A", "B" and "C" indicated increasingly wider tires."A" was originally a tire about 30 mm wide, so the 650A rim is pretty large, 590 mm. If you add the top and bottom 30 mm tire thickness to 590, you wind up with the 650 mm tire diameter.
The 650C size was originally intended for a quite wide tire, about 40 mm wide. Top and bottom 40 mm tire plus the 571 mm rim size again bring you to a 650 mm outside diameter, even though the rim was smaller.