Why was this definition of variance chosen?

Why do we use:

(variance)=(Σ(the difference of each value from the mean)^2)/(number of values)

and do not use instead the:

(Σ(the absolute value of the difference of each value from the mean))/(number of values)?

I guess the because has to do with why the least squares method is the best? Why is the least squares method the best?

I guess the answer to both questions has to do with that the normal distribution has a standard deviation (and thus also variance) equal to 1. So?

Please answer as lucidly as possible (consider I have a low IQ). And explain every mathematical symbolism you use. I'd prefer if you explain it with words alone instead of math symbols.