NVH deals with just what the name implies. Noise, Vibration and Harshness. One area is elimination of road noise and reduction of ambient noise into the interior of the car. This also includes noises introduced by the vehicle itself. Annoyances like dashboard squeek and rattle is a biggie. Transmission/powertrain noise/shifting also plays a big part.
One major test that I have been a part of was "four poster" testing. This puts each wheel on a hydraulic actuator and puts the vehicle thru different profiles of induced shocks. This looks into overall modal issues with frames and suspension systems as well as durability.
It is a pretty diverse area with a lot of responsibilities and a lot of vehicle testing is done, even after production begins.