The most common kind of deviation is the standard deviation.
$$ \text{Sd}(x) = \sqrt{\text{Mean}((x - \text{Mean}(x))^2)}$$
The standard deviation is very similar to the mean absolute deviance or
$$ \text{MAD}(x) = \text{Mean}(|x - \text{Mean}(x)|)$$
but is often simpler to calculate or obeys nice algebraic properties.
But there are a lot of other measures of variance. For example, there is the most common absolute deviance from the mean value: $ \text{Mode}(|x - \text{Mean}(x)|)$.
It is not clear to me why I should prefer the standard deviation over other kinds of measures of dispersion. I suppose the simplest answer is that this measure of dispersion is the most highly studied and well known and using other methods of dispersion will confuse people you try to communicate with.
I guess I would have to use these kind of alternative measures for pathological distributions like the Cauchy distribution though.