Why should I prefer the standard deviation over other measures of variance? 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.
 A: You may come up with infinite number of dispersion measures. It's a lost cause to compare the variance to each and everyone of them.
There are two features of variance that are attractive to me. First, it's a smooth function. For instance, the mean absolute deviation is not.
Second, it's one of the central moments:
$$\mu_k=\sum_ip_i(x_i-\mu_1)^k$$
Here $\mu_2$ is a variance.
Being a moment is important, for it defines the distribution when combined with all other moments. Other measures of dispersion are stand-alone metrics.
A: I'm surprised no one has mentioned the really essential property that Variance is additive for independent random variables:
$$\mbox{Var}(a_1X_1+\cdots+a_nX_n)=\sum_{i=1}^na_i^2\mbox{Var}(X_i),$$
and the equally nice linearity properties that covariance shares. This becomes completely intractable without the square inside the expectation in the definition of variance.
As well for Central limit theorem it is variance, and not L1 that gives rise to the CLT. Specifically L1 gives rise to the strong law of large numbers, but not the fluctuations therein.
A: In some sense, a related and deeper question is why do people tend to use the $L_2$ norm instead of the $L_1$ norm or indeed other norms? In the two-dimensional Euclidean vector space, why do people tend to use $\sqrt{x^2+y^2}$ (i.e. $L_2$ norm) as a measure of distance instead of $|x| + |y|$ (i.e. $L_1$ norm)?
How the $L_2$ and $L_1$ norm are related to standard deviation and mean absolute deviation respectively:
Let $x$ be a mean zero random variable and $P$ be a probability measure.
The standard deviation is simply the $L_2$ norm:
$$ \left( \int |x|^2 \;dP \right) ^\frac{1}{2} $$
And the mean absolute deviation is simply the $L_1$ norm.
$$ \left( \int |x| \; dP \right)^ \ $$
