How did policemen come to be called 'cops'?

British: Constable On Patrol.

From the New York City Police Department Museum

WHERE DID THE TERM 'COPS' COME FROM?

When the first police force began to patrol in the summer of 1845, they only badges on their civilian clothing. The badges were 8 pointed stars with the seal of the City at the center and were made of stamped copper. The newspapers of the time referred to the new force as the 'Star Police' but people seeing the shiny copper shields began to call the new force 'Coppers' which was later shortened to 'Cops.' There is also a British police term; Constable On Patrol which may account for the term 'cops' in England as well.

The British police were origanlly called Peelers after the guy who founded them. They were paid 1 penny, which was made of copper, sometimes also called a copper. So that's why, it's to do with how they paid police wages.

The most commonly heard theories trace "cop" or "copper" meaning "police" to copper buttons worn on early police uniforms, or to copper police badges supposedly issued in some cities, but there is no convincing evidence for any of this. Still other theories explain "cop" as an acronym, standing for "Constable On Patrol," "Chief of Police" or other such phrases. But these "acronym" theories bear all the hallmarks of being spurious after-the-fact explanations invented to explain "cop." Among other sticky details is the fact that acronyms were virtually unknown in English before the 20th century, while "cop" itself was well established by the mid-19th century.

To cut to the chase, the police sense of "copper" and "cop" probably comes originally from the Latin word "capere," meaning "to seize," which also gave us "capture." "Cop" as a slang term meaning "to catch, snatch or grab" appeared in English in the 18th century, ironically originally used among thieves -- a "copper" was a street thief. But by the middle of the 19th century, criminals apprehended by the police were said to have themselves been "copped" - caught - by the "coppers" or "cops." And there you have the etiology of "cop." Case, as the cops say, closed.

http://www.word-detective.com/back-d2.html

In New York when the Police Officer's first uniforms came out , they had bright copper buttons. The slang term came about as a result of this.

Actually, there are two theories. The answer above is one of them. The other theory is that "cops" is an abbreviation for "Constables On Patrol"

http://wiki.answers.com/Q/How_did_cops_become_a_name_for_police_officers

I'd forget the acronym - they're usually made up to fit the word. "While commonly believed to be an acronym for Constable On Patrol, the term refers to "one who captures or snatches". This word first appeared in the early 18th century, and can be matched with the word "cap", which has the same meaning and whose etymology can be traced to the Latin word 'capere'." I don't usually copy and paste, but I couldn't improve on this. https://en.wikipedia.org/wiki/List_of_slang_terms_for_police_officers (This is the one I knew previously as the origin.)