A Kelly day is a recurring scheduled day off on a fire rotation that lowers your average weekly hours. It is named after Chicago Mayor Ed Kelly, whose administration is usually credited with the arrangement in the 1930s. The point is not a vacation day. The point is arithmetic: pulling a schedule that runs over the federal overtime threshold back down under it.
That only works if the Kelly day actually removes a shift from your cycle. This is where most explanations stop, and where a lot of firefighters get surprised on payday. A day labeled a Kelly day that does not drop a tour does not change your hours at all.