Statistical variance gives a measure of how the data distributes itself about the mean or expected value. Unlike range that only looks at the extremes, the variance looks at all the data points and then determines their distribution.
Note that the denominator is one less than the sample size in this case.
The concept of variance can be extended to continuous data sets too. In that case, instead of summing up the individual differences from the mean, we need to integrate them. This approach is also useful when the number of data points is very large, like the population of a country.
Variance is extensively used in probability theory, wherein from a given smaller sample set, more generalized conclusions need to be drawn. This is because variance gives us an idea about the distribution of data around the mean, and thus from this distribution, we can work out where we can expect an unknown data point.