A Reeves drive is a mechanical variable speed system that uses a lever on the front of the lathe to adjust the speed. When the lever is turned, the speed is increased or decreased because there is a special pulley inside that can spread apart or cinch together to change the pulley ratio between the lathe and the spindle.
Usually, there are two belts with the system- one from the motor to the Reeves pulley, and one from the Reeves to the drive spindle. The drive spindle often has a few different pulley sizes so that you can change the overall minimum and maximum speed to suit your preference.
As far as testing it goes, I'd imagine I'd just try it out under power to make sure it's working properly. Be advised that you won't be able to easily change the position of the lever when the lathe is not running- so don't just try it out on a floor model, and assume it is broken!
Personally, I really like mine. The newer lathes usually have electronic variable speed, but I know that if the Reeves drive goes out, I at least have a fighting chance of fixing it myself.