Burns' Flying Monkeys were a division of monkeys that were intended to fly.
While Homer Simpson and Mindy Simmons were staying at the Capital City Plaza Hotel, they ordered room service. This set off an alarm at the Springfield Nuclear Power Plant. Charles Montgomery Burns retorted by sending his monkeys to fly out the window, they failed to fly and fell to their apparent death. Mr. Burns and Waylon Smithers, Jr. suggested more research.
This is a literal reference of The Wizard of Oz. In the film a witch does exactly as Mr. Burns, but they successfully fly.