In more detail
It divides the difference by its standard error and compares the result with the t distribution, which has heavier tails than the normal for small samples. The output is a t statistic and a p-value. A one-sample test compares a mean with a fixed value, an independent two-sample test compares two groups (Welch's version does not assume equal variances), and a paired test compares before-and-after measurements on the same subjects. Degrees of freedom are n − 1 for one sample or for pairs.
Why it matters
It is the standard check for small experiments: does a new process change the average, do two groups differ? Using normal (z) critical values on small samples overstates significance. Treating paired data as two independent groups throws away the pairing and usually hides a real effect. It assumes roughly normal data or a reasonably large sample.
Example
Eight bags labelled 500 g weigh 498, 502, 497, 495, 499, 501, 496 and 498 g:
mean = 498.25 s = 2.375 standard error = 2.375 / √8 = 0.840
t = (498.25 − 500) / 0.840 = −2.08 df = 7
two-sided p = 0.0757 critical t (95%, 7 df) = 2.365
The bags average 1.75 g light, but with eight bags that is not significant at the 5% level. Treating t = −2.08 as a z score would give p ≈ 0.037 and the opposite conclusion.
Common confusion
A z-score uses a known population standard deviation; the t-test estimates it from the sample, which is why its critical values are larger.