For humans:
Firstly, how to move, dodge (not dancing, just how the dodge command works), shoot, buy weapons, use medkit, egg hitboxes, how poison effects you, and how to win/loose a game. The next most important thing to demo IMO is to explain basic concepts of building, since poor building is a great way to make folks dislike you.
For aliens:
How to move, wallwalk, heal (and what effects healing speed), use poison, and the importance of locational damage (headshots!). Then, if possible (and in a separate demo), basic concepts of building which should cover the facts that trappers have a conical range, don't build them on the floor, while tubes have a spherical range, don't build them on the ceiling.
Note: MG tried this at one point, though it didn't really get off of the ground. This isn't a particularly simple task, and I doubt it will work well unless you have all of the onscreen messages scripted before trying to make the demo.