Here is the full list of all 13 The Sun and Moon achievements.
Beat every level in the fourth system
Beat every level in the second system
Beat every level in the first system
Beat every level in the third system
Beat every level in the fifth system
Beat every level in the sixth system
Beat every level in the seventh system
Beat every level in the eighth system
Beat every level in the ninth system
Beat every level in the tenth system
Obtain 100 crescent moon medals
Obtain 100 full moon medals
Obtain 100 sun medals