One of the major categories in statistical tests is parametric and non-parametric. Parametric means that you can describe your data with some commonly used parameters, particularly the mean and standard deviation. To describe your data (more precisely, the distribution of your data) with the mean and standard deviation, you must be able to assume that the population forms the normal distribution. There are additional requirements for some of the parametric tests, but the assumption of the normality is the most important assumption.
The point of the normality assumption is that you can assume that the population from which you take the samples forms the normal distribution, but your samples don't necessarily form the normal distribution. So, even if your data don't look like forming the normal distribution but you can assume the normality for the population, you can use a parametric test. In many cases, you can assume the normality for the population on the observations you have in HCI research, so you will use parametric test. If you want to be more careful about this, you can do some statistical tests for checking the normality (e.g., one-sample Kolmogorov-Smirnov test). Drawing a histogram is also a good idea to see what the data look like.
So, when do you want to use non-parametric tests. Roughly speaking, there are two cases in which you want to use non-parametric test:
It seems kind of rare to see non-parametric test used in HCI papers. Fortunately, many of the parametric tests are fairly robust against the non-normality, so you can try parametric tests unless you think you really need to do non-parametric tests. There are also some ways (e.g., data transformation) to allow you to use parametric test with your non-normal data. You can find more details about statistical tests for checking the normality and data transformation in a separate page.