A "Kelly day" is the occasional day off firefighters get to control budget-destroying overtime. The origin of the name is debated, but an often-repeated tale is of a Chicago mayor named Kelly who gave the firemen one day off after seven on duty, according to Chicago Firefighters Union Local 2.
Firefighters typically work a rotation with extended hours, such as one 24-hour day on and two off, using three teams. With the creation of labor laws in the 1930s and 1940s and standardized pay periods, this meant that one team had an extra shift during a 28-day pay period. Work in excess of 212 hours in 28 days is overtime, so that team either has to have a full shift off or be paid for overtime for it. This off-shift is the Kelly day. Departments with four teams don't need a Kelly day but may still use the term for off days.