In more detail
The curve is symmetric about the mean, so mean, median and mode coincide, and its spread is set by the standard deviation. About 68.27%, 95.45% and 99.73% of values fall within 1, 2 and 3 standard deviations of the mean, and exactly 95% within 1.96. Converting a value to a z-score maps any normal distribution onto the standard normal (mean 0, SD 1), whose table answers every probability question. Averages of many independent values tend towards a normal shape whatever the original distribution (the central limit theorem).
Why it matters
Confidence intervals, z-tests, quality-control limits and many grading curves assume normality. When the data are skewed, as incomes, waiting times and file sizes usually are, the bell-curve percentages can be badly wrong, especially in the tails where rare events live.
Example
Adult heights with mean 175 cm and SD 7 cm:
P(168 to 182 cm) = P(−1 < z < 1) = 68.27%
P(over 190 cm) = z = (190 − 175) / 7 = 2.14 → 1.61%
middle 95% = 175 ± 1.96 × 7 = 161.3 to 188.7 cm
Common confusion
The central limit theorem is about averages, not raw data: the mean of many incomes is roughly normal across samples, but incomes themselves are right-skewed, which is why their median is quoted.