Here is the full list of all 23 Gems of Magic: Lost Family achievements.
Complete the level at last move
Buy at least one booster
Complete the level with a half moves
Create 10 bonuses on a field during level
Destroy 28 gems in one move
Create 5 bonuses on a field during level
Destroy 20 gems in one move
Destroy 12 gems in one move
Use any booster one times
Fail at least one level
1 Hour of gameplay
All moves in level according to the task
Buy all boosters
2 Hours of gameplay
3 Hours of gameplay
You rescued Father!
You rescued Mother!
You rescued Grandma!
You rescued Grandpa!
You rescued Sister!
Create 17 bonuses on a field during level
Use any boosters 7 times
Use any boosters 10 times