As CheekyChops state, a post-graduate qualification is one which comes after a degree. It's also something which can only be obtained by getting a degree first.
'Bachelor' qualifications (e.g. Bachelor of Arts, Bachelor of Science, Bachelor of Eductation) are 'first' degrees. (Study for these qualifications typically takes 3 or 4 years). Someone remaining at university can then go on to take a post-graduate degree. This will normally be a Masters degree. (e.g. M.Sc, M.A., etc). This takes a further one or two years. If the same person continues further at unversity, their next post-graduate qualification will be a research degree, i.e. a Doctorate such as PhD or DSc.
The above examples relate to post-graduate qualifications within universities. Individual professions (such as medicine or the law), however, provide post-graduate qualifications which are not actually degrees but, through membership of professional bodies, may still add further letters after someone's name.
Chris