Police Headquarters (Immigration Office)
Plain English
It's the State Police headquarters in each Italian province. The bit you actually care about is inside, called the Ufficio Immigrazione. That's the office that processes your residence permit, does your fingerprints, and hands you the finished card.
Why it matters
Pretty much every non-EU resident in Italy ends up at the Questura at least once per renewal. Fingerprints, document checks, and permit pickup all happen here. It also handles asylum requests and parts of the citizenship process.
Real example
"Two months after Joseph dropped off his kit postale, he got an SMS giving him an appointment at the Questura di Firenze. He brought his passport, four passport photos, and the post-office receipt. Fingerprints took 15 minutes."
Common confusion
People mix up the Questura with the Prefettura. The Questura is the police side and issues your permit. The Prefettura is a separate civilian office that handles things like citizenship and nulla osta authorisations.
Official source
Polizia di Stato (Questure)