PGY stands for "postgraduate year." When doctors finish medical school, they still need more training before they can practice medicine on their own. This training is called a residency, and it usually lasts between 3 and 7 years depending on the type of doctor they want to be.
During residency, doctors are called "residents" and are assigned to different hospitals or clinics where they work with other doctors to care for patients. Each year of residency is called a "PGY," or postgraduate year.
So, for example, if someone finishes medical school and enters a three-year residency program in internal medicine, they would be called a PGY-1 during their first year of residency, a PGY-2 during their second year, and a PGY-3 during their final year. The higher the PGY number, the more experienced the resident is.