The term "fluorescence" refers to the process by which atoms absorb photons of one wavelength and emits photons at a longer wavelength. A common application of this effect is in nearly every household around the world: fluorescent lights. Fluorescent lights were first introduced in 1939 at the World's Fair in New York City by Interelectric corporation and offered to the public in 1940. An electric current passes through an elongated bulb, colliding with mercury atoms. The collision process excites the mercury atoms, which then emits ultra-violet (UV) light. This emission is actually referred to as "luminescence" or "scintillation". These UV photons are then absorbed by the phosphor coating of the bulbs, which re-emits in the visible. It is of course the re-emission process which is properly called "fluorescence".
The passage of charged particles in an extensive air shower through the atmosphere results in the ionization and excitation of the gas molecules (mostly nitrogen). Some of this excitation energy is emitted in the form of visible and UV radiation.
Rigorously speaking, this is a "luminescence" process analogous to the emission by mercury in a fluorescent light. Much to the horror of the optical physicists, the name "Air Fluorescence" has been adopted by the astrophysics community to describe the scintillation light from extensive air showers. This misuse of the term is in part due to the apparent similarity to the workings of a fluorescent light. On the positive side, this usage makes it easy to distinguish between a fluorescence detector from a scintillation detector (the latter is the name commonly used for desk-top particle detection devices made from inorganic salts or organic plastics).