What is the difference between parametric and nonparametric statistical tests?

Answer 1

Parametric are the usual tests you learn about.

Non-parametric tests are used when something is very "wrong" with your data--usually that they are very non-normally distributed, or N is very small. There are a variety of ways of approaching non-parametric statistics; often they involve either rank-ordering the data, or "Monte-Carlo" random sampling or exhaustive sampling from the data set.

The whole idea with non-parametrics is that since you can't assume that the usual distribution holds (e.g., the X² distribution for the X² test, normal distribution for t-test, etc.), you use the calculated statistic but apply a new test to it based only on the data set itself.