A "Statistic" is a measure of a property of a Sample, for example the Sample Mean or Sample Standard Deviation. The corresponding term for a Population or Process is "Parameter".
The most commonly used statistical tests are "Parametric", that is, they require that one or more Parameters meet certain conditions or "assumptions". Most frequently, the assumption is that the Distribution of the Population or Process is roughly Normal. Roughly equal Variance is also a common assumption.
If these conditions are not met, the Parametric test cannot be used, and a Nonparametric test must be used instead. This table shows the Nonparametric test that can be used in place of several common Parametric tests.
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.