Continuity mistake: When Jake is explaining why he let Richmond go, the left zip of his hoodie goes from folded out to flat between shots and back again a few times. This is especially noticeable when he says "the old me." (00:18:07)
Continuity mistake: When the lights go off after attaching the belt to the ceiling, Holt is interrupted before he finishes the word "winner", however in the flashback he says "winner of the 5th annual-" before being interrupted.
Continuity mistake: When Adrian stamps "denied" onto Boyle's insurance claim form, there's blank spots where there wasn't enough ink on the stamp (which would happen because he had the stamp in his pocket for a while). But then when he says "denied" and holds up the form, "denied" is all filled in fully. Also, the red border is lighter and the location has shifted down.
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.