The simplest answer is that parametric statistics are based on
numerical data from which descriptive statistics can be calculated,
while non-parametric statistics are based on categorical data.
Takes two example questions: 1) Do men live longer than women,
and 2), are men or women more likely to be statisticians. In the
first example, you can calculate the average life span of both men
and women and then compare the two averages. This is a parametric
test. But in the second, you cannot calculate an average between
"man" and "woman" or between "statistician" or "non-statistician."
As there is no numerical data to work with, this would be a
non-parametric test.
The difference is vitally important. Because inferential
statistics require numerical data, it is possible to estimate how
accurate a parametric test on a sample is compared to the relevant
population. However, it is not possible to make this estimation
with non-parametric statistics. So while non-parametric tests are
still used in many studies, they are often regarded as less
conclusive than parametric statistics.
However, the ability to generalize sample results to a
population is based on more than just inferential statistics. With
careful adherence to accepted random sampling, sample size, and
data collection conventions, non-parametric results can still be
generalizable. It is just that the accuracy of that generalization
can not be statistically verified.