What is a "professional student"?
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.
My experience with the term is frequently derogatory. It is usually applied to a person who either doesn't know what they want to study, switches their major every year or two, refuses to graduate, and spends many more years than the average in school before graduating. Or, slightly less bad, is someone who likes school so much that they keep finding ways to get degree after degree after degree, never getting a job related to one of their majors, and typically working menial jobs to keep funding their education habit.