Sampling Distribution
A sampling distribution is the probability distribution of a statistic (a sample mean, a sample proportion, a regression coefficient) over all the samples that could be drawn from a population using a given sampling method and sample size. It is a thought experiment made precise: imagine drawing sample after sample, computing the same statistic from each, and collecting the results; the pattern those values form is the sampling distribution. It is not the distribution of the data in any one sample, and it is not the population's distribution; it is the distribution of an estimator's possible values, which is why it is the central object connecting what was observed in one sample to what can be said about the population.
The smallest example shows how it works. Take a population of three values, 2, 4 and 6, and draw samples of 2 with replacement: there are 9 equally likely ordered samples, with means 2, 3, 4, 3, 4, 5, 4, 5 and 6. The sample mean therefore equals 2 or 6 with probability 1/9 each, 3 or 5 with probability 2/9 each and 4 with probability 3/9. Its average is 4, exactly the population mean (unbiasedness), while its standard deviation is about 1.15, versus 1.63 for the population itself: the means are more tightly clustered than the individual values. That ratio, 1.63 divided by √2, is the general rule that the sampling distribution of a mean has standard deviation σ/√n, which is the definition of the standard-error.
The central-limit-theorem explains why the sampling distribution of a mean is so tractable: for reasonably large n it is approximately normal, centered on the population mean with spread σ/√n, almost regardless of the population's own shape. That approximation is what justifies z-based and t-based confidence-interval and hypothesis-testing procedures, and the standard-normal-distribution or student-t-distribution then serve as the reference. Other statistics have their own sampling distributions: the sample variance of normal data is related to the chi-square-distribution, a ratio of two variances to the F distribution, and the sample median, a correlation or a maximum have shapes that are usually derived by approximation or computed by simulation instead.
When a formula isn't available, the sampling distribution can be approximated by computer. Bradley Efron's bootstrap (1979) treats the observed sample as a stand-in for the population, resamples it with replacement thousands of times, recomputes the statistic each time and uses the spread of those values as the estimated sampling distribution. Permutation tests build a null sampling distribution by reshuffling labels, and Monte Carlo simulation does the same from an assumed model. Data scientists use these tools to put error bars on statistics that have no neat formula, such as a median, an AUC, a ratio of means or a quantile of a latency distribution, and the same logic is behind bagging and random forests, which average over resampled datasets.
The common confusions are between three different distributions: the population, the single sample and the sampling distribution. A histogram of one sample's values estimates the population's shape; it says nothing directly about how a statistic would vary from sample to sample. Another error is believing the sampling distribution is something observed: in a real study there is only one sample, so the sampling distribution is a theoretical, simulated or bootstrap-based construction, and it describes only randomness from sampling. It does not account for bias from a flawed frame or measurement error, so a statistic with a razor-thin sampling distribution can still be centered on the wrong value.
Read a sampling distribution as the answer to the question what else could this sample have said: the full spread of values a statistic would take across every sample that might have been drawn, whose center reveals bias, whose width is the standard error, and whose shape (often normal, thanks to the central limit theorem) licenses every confidence interval and test.