Tutoring, student does not pay I work as a post-doc and do 1-2 hours of tutoring each week (approved by the university), to keep on top on my mathematical and pedagogical skills. Usually, students pay me after each session, but one of my recent students did not pay me the first, or even second session. After that, I have not managed to get in touch with the student again (via email). 

Of course, something could have happened, but I believe I am being cheated. What is proper action here? Should I contact the professor for the course, or perhaps the students main department? Or just forget about it?
This is a private business transaction between you and the student.  Neither the university nor the professor was a party to that transaction, so they'll have no interest in interfering, and it would be inappropriate to try to involve them.

Anyone who provides services to clients for money will eventually find that some of them don't pay.  You can see many examples, and suggestions for ways to collect, on Freelancing.SE.  To say more here would be off-topic, I think.  I'd just keep in mind that pursuing your non-paying client will require more of your valuable time - make sure it doesn't become disproportionate to the amount of money at stake.  At some point, the most cost-effective approach may be to just let it go.