What is a "professional student"? What is a professional student, in the context of a "Graduate and Professional Student Organization" in the United States (cf. this one)? How are they different than or similar to post-docs?
While Bill Barth's description of "professional student" as someone whose sheltering in academia and trying not to graduate is one I have heard, there's also a more formal designation for the term.

Mainly, post-graduate students who are in professional degree programs. For example, medical and dental students, law students, many students pursuing Masters of Public Health degrees, or MBAs, etc. are all pursuing degrees that are intended to be applied to a profession, rather than going further into academia. They often have somewhat different concerns than graduate students, hence the different term.