Here is the full list of all 15 Pixel Space Battles achievements. It takes around 0-1 hour to unlock all of the achievements on Windows.
Complete 10 level
Complete 9 level
Complete 8 level
Complete 7 level
Complete 6 level
Complete 5 level
Complete 4 level
Destroy any enemy on 4 level
Complete 3 level
Complete 2 level
Complete 1 level
Destroy any enemy on 3 level
Destroy any enemy on first level
Destroy any enemy on second level
Start the game