the standard error of the mean
A single measurement scatters a lot, but the average of many measurements is far steadier, the way a crowd's average guess is often better than any one person's. The standard error of the mean quantifies exactly how much steadier: it tells you how tightly the average of your sample pins down the true underlying mean, as opposed to how widely the individual readings scatter.
If the individual measurements have standard deviation sigma (the spread of the raw data), then the mean of N independent measurements has an uncertainty of sigma / sqrt(N). In practice you do not know sigma exactly and estimate it from the sample standard deviation s, giving SEM = s / sqrt(N). The essential content is the 1 / sqrt(N) scaling: to halve the uncertainty on your average you need four times as many measurements, and to improve it tenfold you need a hundredfold more data. Precision is bought expensively.
This is the everyday workhorse that converts the scatter of repeated readings into the error bar you quote on their average, and the same 1 / sqrt(N) law governs the convergence of Monte Carlo estimates. The crucial caveat is not to confuse the standard error with the standard deviation. The standard deviation sigma describes the spread of individual data points and does not shrink as you take more data; the standard error sigma / sqrt(N) describes the uncertainty of the mean and does shrink. Reporting one where you mean the other is a classic error, and the formula assumes the measurements are independent.
One hundred timing measurements of a pendulum's period scatter with a sample standard deviation s = 0.2 s; the standard error of the mean is 0.2 / sqrt(100) = 0.02 s, so you report the period as (its mean) +/- 0.02 s, ten times tighter than any single reading.
The mean of 100 readings is ten times more precise than one reading.
The standard deviation measures the spread of the data and stays put as you add points; the standard error measures the precision of the mean and falls as 1 / sqrt(N). Reporting the smaller where you mean the larger, or vice versa, is a frequent and consequential slip.