Blizzard has replaced their original SC2 demo for a new "Starter Edition" that makes it a little more of a watered down version of the full game. This version is completely free to play for as long as you want.
You can learn about it here:
LinkThe basic limitations are this:
1. Only the first four missions in Single Player
2. Custom games are limited to few maps (these maps will be rotated for other maps during patch changes).
3. Multiplayer is available but only to extremely few custom maps (which are 1v1 maps from the same map pool as custom games).
4. You are restricted to playing the Terran race only. You may still fight against Zerg or Protoss players (these players own the full game and are able to play against those with the starter edition).
5. You do earn achievements like you normally would as if you did own the game for the portions you can play. These achievements will transfer over if you do decide to buy the full game and use the same battle.net account you used in the starter edition.
And that's basically it. More detailed info can be found in the link above. So enjoy some StarCraft for free.
It is not known if the trainer will work with the starter edition but you are more than welcome to try.