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?