Standard deviation measures spread. It is the typical distance of a data point from the mean, expressed in the same units as the data — which is why it is preferred over variance, whose units are squared and therefore hard to interpret.
Why n − 1 for a sample
Using the sample mean rather than the true population mean systematically underestimates the spread, because the sample mean sits closer to your particular data points than the real mean does. Dividing by n − 1 instead of n corrects this bias. Use n only when your data genuinely is the entire population — every student in one class, not a sample of students.
Reading a standard deviation
For roughly normal data, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. This is the empirical rule, and it is what makes a standard deviation intuitively meaningful. It fails badly for skewed distributions such as income, where the median is a more honest summary.
Sensitivity to outliers
Because deviations are squared, a single extreme value can dominate the result. Median and interquartile range are the robust alternatives when your data has genuine outliers you cannot justify removing.