Use of the term "cop" to refer to a police officer came about because a "copper" is someone who captures or seizes something in English slang. The word "cop" came to mean apprehension in the mid-19th century, and since police officers often capture or seize criminals, English officers came to be known as "coppers." The tradition spread to the United States, and eventually it was shortened to "cop."
There are a number of apocryphal theories concerning the true origin of the term "cop." One of the most common theories suggests that it is an acronym for "constable on patrol," but there is no historical or etymological evidence to back up this interpretation. Most acronym-based words entered the language in the 20th century, and police officers have been called cops or coppers for much longer. Another common misconception is that the term refers to copper buttons that an officer would wear on his uniform, but this is similarly unsupported by any historical evidence.
Police officers have been called coppers since at least 1846, and by 1859, the shortened term entered common use. Before 1846, police officers in England were referred to as "Bobbies," named after Sir Robert Peel, the creator of the first Metropolitan Police force in London in 1828.