Sclera

The white part of your eye that you see when you look at yourself in the mirror is the front part of the sclera. However, the sclera, a tough, leather-like tissue, also extends around the eye. Just like an eggshell surrounds an egg and gives an egg its shape, the sclera surrounds the eye and gives the eye its shape.
 
The sclera is also attached to the extraocular muscles, which, in turn, move the eye left and right, up and down, and diagonally.