A fur ball, according to the United States Air Force, is a close-range dogfight involving large numbers of airplanes. The best reference I can come up with for this usage is William Safire, writing for the New York Times Magazine in 2006 in which he references a U.S. Air Force dictionary from 1983. The term references the images of 'fur flying' during animal combat, and a 'ball' (in the sense of a social event) as a large, complex gathering. The sinuous paths of airplanes in close combat can be likened to the maneuvers of particularly agile animals intent on doing each other harm, and if two large forces meet in the air it will be difficult to shake similarities to large, organized events which result in twirling motion of the attendees at the center.
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