What's the sense of standard deviation?

It's like a distorted mean of errors, skewed up, and always positive.

I get the sum of all the deviations from the mean, and i get the division by the number of measures, that would be the average error, which seems a more reasonable measure to me, i also get the elevation to square to have all positive numbers, and also the square root to make go back to its previous size, but shouldnt the denominator be elevated to square too? Otherwise it's warped, i don't understand.