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 that's happening in her precinct, she knows to hop in her 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.