=Half-Life: Uplink=
This'll just be a quickie since Uplink is so short, but this was the official Half-Life demo that came out awhile after its release. It featured three brand new levels that used stuff cut out of Half-Life. IMO, it's an excellent demo as it demonstrates all the core things about Half-Life, gives you marines and their excellent AI to fight, and gives you a hell of a closer with the gargantuan bearing down on you.

There's a Gman sighting here, of all places, when the Gargantua is trashing the room. Also, there's a few things introduced in the demo that Gearbox then went on to use in their expansions (Military using torches to get through walls, and gas canisters as explosives). All in all, this was a good way to introduce people into Half-Life, though it was a tad bit hard due to a lack of weapons early on. Such is life.

Once upon time you could download a mod version of uplink through Valve's new mod viewer; but since that was a generation of Half-Life ago, it seems to have dissapeared. I still had it lying around though so lucky me, but I couldn't tell you where you could find it yourself.