Hey lamp23.
The standard deviation is defined to be the square root of the variance. The variances have good properties that make them useful for many applications since they act like norms and inner products (with covariance terms) which gives these things a geometric interpretation.
The standard deviation is mostly useful for normal or other symmetric distributions.
In terms of understanding this in depth, you will need to study decision theory and statistical inference among other things to understand why means and variances are used over other things like medians (which is along the lines of what you are getting at).