Post Graduates are those students who have attended HOL Hogwarts for 7 (or more) years. They qualified for graduation to the next year according to the requirements in effect at the time. They took (and passed) their OWLs in their 5th year (this requirement sometimes extended to their 6th and 7th years). In their 7th year they took and passed their NEWTs.
Some Post Graduates continue to be students here at HOL Hogwarts. Others have become professors.