The commercial part of CDL doesn't really mean shit anymore it is all weight based and has nothing to do with commercial.
Class C CDL is for vehicles that carry 16 or more total occupants including driver.. ever wonder why they stop at 15 passenger vans?? OR hazardous materials. If you had 15 of your own kids and you drove them to school you would technically need a class C CDL even though they are your kids.
Class B CDL is required for anything over 26,001lbs even if its you are just driving your motorhome full of your gold to the grocery store for the hell of it. Even if not over 26Klbs combined trailer is limited do 10K lbs or less(as is your normal drivers license I think.) So if you had a 5Klbs truck and a 10,001lbs trailer this wouldn't be good enough.
Class A CDL is for anything over 26,001lbs combined truck and trailer and/or anything with a trailer 10,001lbs or more.
The labeling of classes even for passenger cars and motorcycles may vary slightly from state to state but they all basically fall in line between states.
As for UPS or Fedex as far as I am aware they do not require any form of CDL for the standard box delivery vans as they are not over 26Klbs and there is no trailer. My neighbor back in NJ was a fedex driver and I recall him telling me one time when a harry potter book came out that they were out pulling over and ticketing every FEDEX truck because they were overweight and the drivers don't have the licenses they required to go along with the weight just normal drivers licenses.
The tests are different for each class and most schools supply you with a truck as part of the price to take the test or sometimes even the employeer that is hiring you would supply it, if you have a job already.