A few theories as to why they're "coppers", gleaned from various sources: 1. From Constable On Patrol in 1800s England 2. From copper badges worn by New York police officers late 1800s/early 1900s 3. From the Latin "capere", meaning "to take". Cop, meaning grab, capture, snatch etc, entered English in the Middle Ages. A criminal can "cop" (swipe, steal) an object from a store, and the police can also grab, or cop him (hence, "it's a fair cop, guv."). If you want to know why they're also called the Old Bill, here's a site:
http://www.met.police.uk/history/oldbill.htm