You have to rotate a magical cube to group the little vizati stones by color and figure out the best way to do it with a limited number of rotations/moves.
In story mode you can follow the story between Julie, Peter and their friends, who discovered the Vizati cube. You have a limited amount of movements by level to solve each puzzle. As you progress you the story will keep continuing as a backdrop even when you are playing.
Arcade is a rawer, faster and even more teasing mode with special stones. Some will freeze you game, others will help you by switching to a needed color. The Arcade mode is sub divided into 2 objective types.