Continuity mistake: The scene where Boyle explains the case for Scully and Hitchcock you can see the same two people walking by the room 3 times, coming from right to left in the background. (00:08:30 - 00:09:20)
Continuity mistake: When Jake and Boyle talk about Jake's Dad, a man (grey clothing and holding a cup) in the background (behind Boyle on the left) walks from the back to the front in the same way in both shots. (01:45:00 - 01:55:00)
Answer: In reality, the Commissioner is a civilian administrator, appointed by the mayor of New York City. He (as of this writing, the role has always been filled by a man; the show gets that part right) is usually someone who has risen through the ranks of the NYPD, but upon becoming Commissioner, is no longer a sworn member of the force (the highest sworn rank is that of Chief). Since Commissioners are appointed, technically anyone can serve in the position, including Deputy Commissioners (who are appointed, in turn, by the Commissioner himself).
Update: On 1 January 2022, Keechant Sewell became the first ever female commissioner of the NYPD.