"The Windy City" is a nickname for Chicago, Illinois. As The Straight Dope explains, the origin of this nickname is not necessarily clear. Some believe that the nickname refers to the fact that the city harbors a strong wind (lovingly referred to as The Hawk) that comes off Lake Michigan and blows between the buildings with a truly chilling effect.
Another theory is that the term came in to common use when Charles Dana, editor of the New York Sun in the 1890's, wrote of Chicago as "...that windy city," referring primarily to the city's excessive boasting and talkative nature. Earlier cites refer to Chicago as "The Windy City", but in these contexts it is not altogether clear what is meant by the expression.
If the latter theory is correct, it would not be the only derogatory nickname Chicago has obtained. Another nickname for Chicago is "The Second City," which I have been told also referred to its supposed inferiority to New York. In both cases the name stuck, and Chicago seems to keep its sense of humor about it.