Best known from the best-known sequence in Walt Disney's Fantasia, The Sorcerer's Apprentice originated as an old folk tale. It was set in poetry by Johann Wolfgang von Goethe as Der Zauberlehrling (The Pupil In Magic). It, in turn, was set to music by Paul Dukas. And that, in turn, was finally set to animation by Disney.
A magician's assistant, having watched his master ply his craft (but not quite closely enough) casts a spell on a broom to fetch water for his chores. He forgets the incantation to stop the broom, which soon overfills the basin and shows no sign of stopping. Desperate, the apprentice attacks the broom with an axe. Naturally (or should I say, supernaturally) the halves each come to life and continue the chore. Water, water, everywhere. Finally, the master calls the broom to a halt.
The term "Sorcerer's Apprentice Mode" is sometimes used to refer to the rapid spawning of processes or sending of messages by out-of-control software.
-- DaveLand