An endocrinologist or ObGyn would probably give you a clinical definition. She might tell you that PCOS is short for PolyCystic Ovary Syndrome and that it has gone by many different names over the years including PCOD (PolyCystic Ovary Disease), PFOD (PolyFollicular Ovarian Disease) and also Stein-Leventhal Syndrome, named for two doctors who first published a description of the syndrome. She might go on to tell you that we've known about PCOS for almost 80 years and that its clinical definition has evolved in that time.
The NIH published criteria in 1990 (PMID: 21263450). A meeting of specialists voted to include a new criterion in 2003. The Androgen Excess Society (AES) published its own criteria in 2006 (PMID: 20186113). As recently as September 2011, a new study strongly suggested the minimum number of follicles per ovary should be revised from 12 up to 19 in order to diagnose PCOS.
So, hopefully, you can see that the exact clinical definition of PCOS is still changing and will continue to evolve as our understanding improves. For now though, PCOS is generally defined as having hyperandrogenism plus polycystic ovaries and/or irregular menstrual cycle. There are several of other health issues that can mimic PCOS, so they need to be ruled out before a diagnosis of PCOS is made.
There are other ways to define and describe PCOS though. For example, you could say that PCOS is the leading cause of female infertility in the US. You could also say PCOS is a major risk factor for numerous other health issues like metabolic syndrome, type II diabetes, fatty liver disease, heart attack, precocious puberty, miscarriage, and premature ovarian failure. All of these would be good answers to the question: What is PCOS?.
And yet even the best answer would still be incomplete. Until we have a clearly defined etiology (or etiologies, since it is quite unlikely that there is just one) for PCOS, any answer we give will be incomplete. But, research continues, our understanding improves, and someday we will know what causes PCOS.