This isn't the do-all, end-all guideline, but roughly:
C Shows: Local barn shows, also called schooling shows. No points earned (unless it's for the hosting barn), but there is a judge, places given, etc. Turn out for horse and rider tends to be more relaxed. Colors (jackets/breeches/shirt) are mixed, you'll see some half chaps with paddock boots, maybe some bright schooling helmets. Entry fees/grounds fees/class fees are very reasonable.
B Shows: State/Regionally rated. Riders accumulate points for end of the year awards in whatever organization. Shows are more proper than 'C' shows, but more forgiving than 'A' shows in terms of turnout. (Which probably depends on the organization/area.) Cost is also higher, but not extravagant.
A Shows: Nationally rated, bigger prize money, but also more fees and higher cost per class involved. Horses and riders are both turned out impeccably. Proper colors, tall boots, black helmets, horses braided, hooves polished, etc.
Some of the details (turnout, costs) may also depend on your area. And of course, if anyone has anything else to add to my definitions, feel free to. :)
If you have never shown or been to a show, I suggest you go and watch some of the jumper and hunter shows they are advertising that have a B and C rating. Then if it looks like fun, you could enter a "schooling"show. They are fun. That is mostly what we do. Every once in a while we go to a "local" show, which I guess is equivalent to a "'B"" show.
TECHNICALLY... a show gets a "A" or "AA" status by how much prize money they give. I think it has to be at least $6500, and cannot take place within a certain range of another A show (like 100 miles) on the same date. But Opus's description was pretty accurate.