Your Honor - S4-E18
Continuity mistake: In the cold open, when Jake throws the door handle at the interrogation room mirror trying to break it, the handle bounces off the glass to his right, but when the shot cuts to in front of him to show his reaction, the handle flies off to his left. (00:01:00)
Chasing Amy - S4-E17
Continuity mistake: The position of the money Charles puts on Gina's desk keeps changing through the scene.
Mr. Santiago - S4-E7
Revealing mistake: When Jake is typing on his computer just before Terry talks to him about meeting Amy's dad for the first time, his computer is and remains on the desktop doing nothing. (00:02:05)
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.