The standard deviation is the most common measure of deviation or variability. When used in conjunction with the normal curve, the standard deviation provides a much clearer picture of variability. See the section, Properties of the Normal Curve, for an explanation of the relationship between the standard deviation and the normal curve. See the graphic, Types of Distributions, for examples that show how the mean and the standard deviation may be related. Mathematically, variance is the square of the standard deviation.
Many statistical tests use variance explicitly.
Multivariate statistics provide the proportion of the variance attributable to a given independent variable. To understand variance, think of decision making. The final decision you make will then be the sum of a number of independent variables, each with its own variance. Some variables will have larger effects than others. For instance, if the new job doubles your pay in real terms, then that variable may affect your decision much more than the fact that your children are upset about leaving old friends.