A precinct is an area of town marked off for official purposes, often to vote, or to organize the police force. When a cop hears of a crime on the police blotter that's happening in his precinct, he knows to hop in his car and head over there.
The word precinct comes from the Latin precinctum meaning "enclosure, boundary line." A precinct is an area that has a clear boundary line around it, making it easier for the police to know where a crime is taking place, or where you should go to vote. The word precinct has also come to mean police headquarters in a particular district. If you're being held at the precinct, you're at the police station local to where you committed your crime.
