Everyone thinks that Chicago is referred to as the Windy City because of the howling northeast winds that can blow across the city off of our Lake Michigan, and yeah--those winds are something else, nasty as can be!! But hey, stick around for a few minutes and the weather just might change. The real meaning to the term is due to the big mouth politicians that reside in and control this city. Though the term was put into use at the turn of the 20th century, it is still so very true today.
The Chicago Historical Society explains it better. The term had to do with the 1893 World's Columbian Exposition, which celebrated "the 400th anniversary of Christopher Columbus's landing in America." The CHS also says:
New York City, Washington, D.C., St. Louis, and Chicago had all vied for the honor of housing the exposition, and it was during this vigorous and often vocal competition that Charles A. Dana, editor of the New York Sun, dubbed Chicago "that windy city." Chicago's lobbyists finally won out and, on April 25, 1890, President Benjamin Harrison signed the act that designated Chicago as the site of the exposition.
If you want to check out the origin of other English idioms, check out the idiom site. I just can't guarantee that all their information is correct.