In statistics, an exact (significance) test is a test where all assumptions that the derivation of the distribution of the test statistic is based on are met. This will result in a significance test that will have a false rejection rate always equal to the significance level of the test. For example an exact test at significance level 5% will in the long run reject true null hypothesis exactly 5% of the time.
Parametric tests are exact tests when the parametric assumptions are fully met, but in practice the use of the term exact (significance) test is reserved for those tests that do not rest on parametric assumptions – non-parametric tests. However, in practice most implementations of non-parametric test software use asymptotical algorithms for obtaining the significance value, which makes the implementation of the test non-exact.
So when the result of a statistical analysis is said to be an “exact test” or an “exact p-value”, it ought to imply that the test is defined without parametric assumptions and evaluated without using approximate algorithms. In principle it could also mean that a parametric test have been employed in a situation where all parametric assumptions are fully met, but it is in most cases impossible to completely prove this in a real world situation. Exceptions when it is certain that parametric tests are exact include tests based on the binomial or Poisson distributions. Sometimes permutation test is used as a synonym for exact test, but although all permutation tests are exact tests, not all exact tests are permutation tests.