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.
Secret
Try the 6th theme
Try the 7th level
I don't know what it is, but it's on your keyboard!
Try the 4th level
Try the 22th theme
No moves left
Use hint three times in the same game
Match 8 tiles
Get at least 10 karma
Match two tiles
Try the 5th level
Try either the 19th theme or the 21th one
Run the game for the first time