Here is the full list of all 13 Neuroshima Hex Demo achievements.
Play any mode at least once
Play and finish 10 singleplayer games
Play and finish 10 online matches
Play with each of the available armies
Win with every available army
Beat the AI on each of 3 difficulty levels (easy, medium, hard)
Win an online match 3 times
Finish tutorial
Get all other achievements
Surrender a game
Lose 3 games
Win 3 matches in a row
Use all hexes