Class A license - certified to drive tractor-trailer trucks.
Class B license - certification to drive non-tractor trailer multi-axle vehicles; kind of an amorphous collection, but think buses, large commercial panel trucks, dump-trucks, highway plow trucks, etc.
Class C license - any kind of normal passenger automobile.
What I find interesting about Uber, and the backlash against it, is that it is largely just a scaling of under-the-table, unregulated behavior people have always indulged in. What, really, is the difference between calling up your circle of friends and acquaintances, asking for a lift somewhere, and throwing them some cash for gas money and their time, versus using Uber? They are evil because they enable you to search a wider network then just your personal circle? Or because they take a cut for providing a reliable, safe, centralized payment service? Or because they have enabled a market that competes with a rent-capturing incumbent, and the product they provide is cheaper, more convenient, and higher quality?
To me requiring a special kind of driver's license to transport passengers makes a fair amount of sense iff it's reasonably easy to get.
Uber doesn't require that for UberX, which is the main point of conflict in many of the trials Uber lost over here.