I never felt that I understood the fuss about the standard deviation, until I encountered Chebyshev's inequality.
Suppose you have a bunch of data. Clearly, the mean of the data is somewhere in the middle, and the data are spread out more or less far from the mean. The standard deviation says how far away they are, in the following sense: no matter how the data are distributed, and no matter how spread out they are, at most $1/n^2$ of them are more than $n$ standard deviations from the mean.
For example, if the standard deviation is 2 units, then all but at most 1/9 of the data are between $m - 6$ and $m + 6$, where $m$ is the mean. Often the data will cluster more closely around the mean than this, but even if you know nothing else, you can guarantee that no more than 1/9 of the data are more than 6 units from $m$.