Symmetry is important because it is the main feature of nature that restricts the permissible dynamic laws. The concept of balance is very important to understanding how symmetry works.
For symmetrical balance, a vertical axis is needed. The symmetry axis for human beings is vertical, which is why balanced weight is needed on both feet. Humans have to find a symmetrical balance in order to learn how to walk or ride a bike.
Symmetry is applicable in classical laws of motion and quantum mechanics. The law of nature is based on symmetry, but with many symmetry-breaking mechanisms in place. One new feature is supersymmetry, an extension of the geometric symmetry of space and time that also includes fermionic or anticommuting charges.