Everything is important! Here are some off the top of my head, but I am sure some of the pros will add to this.
1) Operate all of the appliances and electrical items and make sure they all work correctly. This includes A/C, furnace, hot water heater (on both gas and electricity), fridge (on both gas and electricity) microwave, stove, oven, tv if present, sound system, awning (manual or electric), jacks if electric, lights inside and out, and anything else it might have that performs a function.
2) Make sure the slide(s) operate correctly (in and out) and learn how to manually move the slide (where is the crank and the access).
3) Check all plumbing for leaks and function (run water from all faucets).
4) Check the obvious for workmanship (appearance and condition such as tears or peeling)
5) Check tires, have them explain pressure, lug torque, bearings, etc.
6) Make sure battery is secured.
7) Have them explain propane valve operation and point out all propane devices and outlet valves.
And of course, the overall appearance and condition of the rig.
I am sure others will add to this list.