ANSWER
It was originally used during World War II to keep moisture out of ammunition cases.
The tape we know today as "duct tape" was invented during the war by Permacel, a division of the Johnson & Johnson Company. Because it was strong, waterproof, and easy to use (you could rip a piece off the roll by hand) it was soon used for a variety of repairs to jeeps, aircraft, weapons, and other equipment. At the time it was called "duck tape", both because it was waterproof, and because the cloth used for its middle layer was called "cotton duck". It wasn't until after the war, with the booming housing market and the popularity of forced-air heating systems, that it began to be used to hold together air ducts. The name was changed to "duct tape", and it color was changed from army green to the silver to better match the color of the ductwork. Today "duck tape" is a brand name of Henkel Consumer Adhesives, a company formerly known as Manco, and one of the early post-war manufacturers of duct tape.
In addition to its practical uses, duct tape has spawned a sort of sub-culture of people who like to make clothing out of the stuff. (For more info about duct tape clothing, visit http://www.ducttapefashion.com) Some have even suggested that duct tape was George Lucas' inspiration for "the Force", because it has a light side, a dark side, and it holds the universe together.
WHO GOT IT RIGHT: JP Weigt, Robert Walker, Marika Thiessen, Robin Campbell, Allan Christensen, Jerome Brown, Jim Ahumada, Kristy Morgan, Marc Quinlivan, Bob Milligan, and Yanitz Grinell.