When you want to test the relationship (correlation) between two continuous variables, the main methods are easily learnt and very well documented. Scatterplots with regression lines, pearsons correlation tests and linear regression models reveal a tremendous amount of insight.
However, if you want to generate similarly useful insights about the relationship between a discrete and a categorical variable, information about appropriate methods seems very thin on the ground. Certainly this wasn't covered in my stats courses.
What are the tried and tested ways of exploring relationships between discrete and categorical data?
Best Answer
In this particular case, lets call discrete variables "ordinal", depending on the number of values, as the latter quality would determine how it might be analyzed.
The easy answer is that if you have 7 or more values in the ordinal variable, it starts acting like a quantitative variable and it can start to be analyzed in that fashion. In either case, the graphic to begin with regarding categorical vs ordinal would be a box-and-whisker. (unless there were very few ordinal values)
If the ordinal variable has 7 values or less, you might treat it as a categorical variable. But there are measures specifically for categeorical vs ordinal, like the eta, or the intrepid rank biseral.