To go along with ksjp17's comment, it really boils down to the kind of demo you are releasing, and (as ksjp mentioned) the reason. I'll break down the variations of what I've seen, and my own thoughts:
Tech Demos: These are mainly to generate feedback on ideas during the project development. They are primarily focused on a single type of mechanic within the game, rather than the game itself, so there isn't going to be any story plot, nor will the player be able to play all of the features the game has to offer. The bits of dialogue their are will either be from an in-game character or the narration of the developer to inform the player about the technique they are using, but this is not necessary. Most times, these last less than 15 minutes (usually around the 5 to 10 minute mark), but seldom over 15 minutes.
Most times this isn't even a scene in the game, so a developer will just mock up a quick project with this feature and see what people think.
Developing Demos: Not sure if there is a proper name to give this, but this occurs when a game is in development stages, and the developer wishes to release their demo to the audience before the game is even finished. The sole purpose is usually to get feedback in order to fuel the rest of the project, and to make quick changes (if necessary) based on the reaction of the audience. Be careful here, though, because it's easy to get sucked into making change after change rather than actually finishing the game. If you go with this, I say go with the 80/20 rule. Listen to what the majority of your audience is telling you, and decide what changes to make (if any) based on that. While these demos can last a good while, generally they cover 1/4 to about 1/2 of what the complete game would be.
Where it starts is anybody's choice. Most start it at the beginning, with the knowledge that the completed game may be changed or drastically altered before it's finished. But since you want the best amount of feedback and enthusiasm as you can get, some may decide to start it somewhere in the middle. Be careful of this, though, and think about the reasons behind starting the game in the middle. If the focus on the demo is a portion of the story or mechanic you want to show off, or to get players immediately into the action, I say go for it. But if you're not happy with how the first part plays out or feel it may alienate your audience, you're not helping yourself so much as setting up false expectations, in which case the feedback you get from showing your weakest part may be more beneficial.
Alphas and Betas: These are usually handled by playtesters (not open to the public unless you want your public to all be playtesters) to find bugs and test the game. Usually reserved for near completed games, although some understanding is expected by the audience that additional development may be needed.
Timed and Area Demos: These are usually reserved for commercial games and are used to entice the player to pay money for the full game. It's best to do this when the game is completed and ready for purchase. 1 to 2 hours is all that's usually needed, although some developers don't put a time limit in and instead have the demo locked when they get to a certain point. In this case, I say have the player be able to play from the beginning of the game rather than in the middle, so that they can easily carry their save file into the full version if they decide to buy it.
