Here is the full list of all 16 That's Mahjong! achievements. It takes around 0-1 hour to unlock all of the achievements on Windows.
No moves left
Match 8 tiles
Get at least 10 karma
Match two tiles
Secret
Try the 22th theme
Try either the 19th theme or the 21th one
I don't know what it is, but it's on your keyboard!
Try the 7th level
Try the 6th theme
Try the 5th level
Use hint three times in the same game
Try the 4th level
Run the game for the first time