Very recently, I tried the demos for both Braid and The Longest Journey, both of which are on Steam.
Braid's demo was great - it gave you all the story nuggets you usually get in the main game, then proceeded to give you the entire first world, half of the second, and one or two levels of the third. The first two worlds gave me a good idea what the game contained, and the third was the 'look what we can do!' hook that got me to buy it.
The Longest Journey was a little odd, since it does actually dump you in the middle of the story and doesn't explain anything to you. You're plonked on a boat at sail, you just have some weird 'Amulet of the Balance' (plus some candy), and your character is keen to get to some island to save the world. Somehow. And the captain is adamant not to go there so to avoid a Chaos Storm. Or something.
Yet regardless of not knowing the full story, I DID manage to relate to the scenario of trying to get this boat to go towards the island I wanted to visit. The demo lasts the entirety of the boat scenario, meaning you don't get cut out whilst in the middle of doing your thing. The game lets you finish off this little thread of the story, leaving you wanting to see what your efforts ended up doing in the terms of the full story.