When someone refers to "brass monkey" weather, he or she is using an English idiom to indicate that the weather is extremely cold. This is the type of idiom which is used in the most extreme circumstances to add some expressiveness to a basic description. The longer version of the phrase is used when someone says, "It's cold enough to freeze the balls off a brass monkey." It is difficult to ascertain whether the phrase had nautical origins referring to the way cannonballs were stacked on a ship, but most who use it in modern times understand that the word "balls" is sometimes slang for testicles.
Idioms are short phrases that have gained meanings over time that are often quite different than the literal definitions of the words they contain. Their meanings often slowly change once they are originated, to the point where they are often used as a kind of colorful shorthand to describe something that might not seem to have any connection with the idioms. Some idioms are extremely colorful in nature and might even be considered a bit risque. One of these bawdily humorous idioms is the phrase "brass monkey" to describe weather.
The meaning of this idiom is simple enough. When weather is described in this manner, it means that it is very cold outside. Usually, this description is saved for cold weather that is extremely out of the ordinary. It allows the speaker to make light of the frigid weather. As an example, a person might say, "According to the forecast, the winds are going to be so cold tonight that it will be real brass monkey weather."
While the accepted meaning of the phrase is pretty simple, the origins of the phrase are a bit more complicated. The phrase is often elongated into the complete sentence, "It's cold enough to freeze the balls off a brass monkey." In this respect, it would seem to mean that the cold weather actually does damage to the testicles of an inanimate object.
There are some explanations to the origins of this phrase that have come up since it first gained some momentum in the first half of the 20th century. The most common explanation refers to the alleged practice of enclosing cannonballs on a ship in a brass device, which, if it contracted from the cold, would allow the cannonballs to fall. This account, however, doesn't seem to gibe with the timing or usage of the phrase. Instead, it is most likely that "brass monkey" was just intended as a humorous idiom to be used in extreme cold.