Asymmetry says something about the symmetry of an object. When an object has no symmetry an object would be asymetrical. In our daily lives we are very familiar with symmetry; our body is (almost) symmetrical along the vertical axis. When we stand in front of a mirror, we form a symmetrical relationship with our mirror image. If you could split a shape in half and you would compare the left side and the right side, or the above and the below, they would not look as each others reflection.
By the way, it does not mean that both sides are identical. For example, your mirror image does not look exactly the same as what other people see (unless you're a supermodel with a super symmetrical look). It means that your image in the mirror is reversed compared to yourself.