Standard Deviation is a measure of variation (or variability) that indicates the typical distance between the scores of a distribution and the mean. Looking at an example will help us make sense of this.
Assume a professor is interested in the satisfaction of students in her psychology class. She decides to survey the students by asking them to rate the class from one to five. From the surveys, she calculates the average score to be three. From this she can assume that people's satisfaction was average. Wanting to know more she decides to calculate the standard deviation and finds it to be equal to two--meaning, the amount of variability between the numbers was 2. This means that most scores were either a one or a five (thus producing the average of three), showing that students were either very satisfied with her class or very dissatisfied with her class (they gave ratings of 1 or 5 most frequently). By obtaining a measure of variability, she was able to understand more about how people felt with the class than she would of with just an average score. This is one of the reasons why standard deviation (and variability) is so important.