The goal is to remove every shape. You have a limited number of tries on each level. Getting shapes of the same color to touch will allow you to remove them all. Shapes that have diagonal lines/twills across them are fixed and static. They are not affected by gravity and will not move. Normal shapes (with a solid fill) are affected by gravity and will move. Shapes containing arrows use a different direction of gravity, as they will fall in the direction of the arrow.
Just click on any shape to destroy it. The goal to clean levels in minimal number of clicks. If more shapes with same color have been connected, you can destroy them in one click.