In more detail
To compute it, take each value's distance from the mean, square it, average the squares (that is the Variance) and take the square root. Dividing by n gives the population standard deviation, used when you have every value; dividing by n − 1 gives the sample standard deviation, used when the data are a sample from something larger, because a sample's own mean sits slightly too close to its values. Roughly 68% of bell-shaped data lies within one standard deviation of the mean. See How Standard Deviation Works.
Why it matters
It tells you whether an average is typical: two classes with the same mean of 70 are very different if one has SD 3 and the other SD 15. It is the yardstick for z-scores, control charts, investment volatility and error bars. Tools differ in which version they default to: Python's statistics.pstdev and Excel's STDEV.P use n, while statistics.stdev and STDEV.S use n − 1.
Example
Data 2, 4, 4, 4, 5, 5, 7, 9 (mean 5):
deviations −3 −1 −1 −1 0 0 2 4
squared 9 1 1 1 0 0 4 16 sum = 32
population SD = √(32 / 8) = √4 = 2
sample SD = √(32 / 7) = √4.571 = 2.138
Common confusion
The standard error, SD / √n, measures how precisely the mean is known, not how spread out the values are: here 2.138 / √8 = 0.756. Error bars must say which one they show.