Below you will find some files that I made for students to help them understand interaction effects, including the regression coefficients that result. The first is an example for a relatively simple scenario with categorical variables and a two-way interaction. The second is a more complicated one where one of the categorical variables has three levels and also there is a three-way interaction effect. Both examples are based on simulated data that are provided in the accompanying rcode; the two-way interaction materials were adapted from the three-way interaction materials.
Note: Something important to keep in mind with interaction effects is that they tend to require a lot larger sample size than main effects, to have some power. Andrew Gelman has a nice blogpost about it.
I made this after a student approached me and asked me if she "could" do three-way interaction effects for her master thesis (not with me as a superviser, but in the context of a stats class) given that everyone always taught her that three-way-interactions just were too complicated; but, as she indicated, it followed so naturally from her research question. I indicated she could definitely do it given that it made a lot of sense in the context of her question, and she did not have particular issues interpreting it. Power is however a concern (see the note towards the top of this page).
In any case, somehow threeway (and up) interactions have achieved a mythical 'too-hard-to-think-about' status. I dislike that, so I worked out an example to share with the students in that class, to help them understand, and now I'm sharing them with you.