Measuring spread, not just average
Two data sets can share the same mean but look very different. Standard deviation captures that difference by measuring how far the values typically fall from the average. A small standard deviation points to consistency, while a large one signals high variability.
Sample versus population
The choice between dividing by n or n - 1 depends on whether your numbers represent an entire population or a sample of it. Dividing by n - 1 corrects for the tendency of a sample to underestimate the true spread, which is why sample standard deviation is the default in most real-world analysis.