Not to discourage new answers, but I've written about medieval and early modern methods of maintaining order in a few previous answers:

Law and its applications to commoners and nobles

Were "guards" more like modern American police, or security guards?

To expand slightly: towns policed themselves, in that it was generally the citizenry acting with an elected or appointed officer of the peace - a sheriff or constable - to apprehend criminals when necessary. It was necessarily reactive rather than modern proactive police forces, which are, in theory, meant to deter crime as well as respond to it.

Larger cities might also have professional or semi-professional agents to handle common crimes, like thieftakers or bounty hunters, but this would be different from one city to another. Sources in the linked answers above.