How do phosphors emit visible light when X-ray is incident on it? I don't exactly understand how phosphors work. Why does one X ray photon produce several visible light photons?
I'd appreciate it if someone explained this to me.
 A: There's a short answer: light (including X-rays)
interacts by its electric field with charges, i.e. electrons.   That interaction is two way; light can activate electrons, and active electrons can create light.
Most X-ray absorption is a photoelectric effect, in which an electron is freed from its atom, to either escape
the material (the Auger effect) or to knock loose
other electrons, or move electrons to higher energy stable orbitals of the material/molecule/atom.
An X-ray photon will impart several keV of energy.
By one rule-of-thumb, however, a single 3keV
electron will likely generate a cascade of
electron-electron collisions resulting in a hundred
or so lower-energy electrons.
The quantum mechanical states for electrons in
any given material generate sharp
spectrum lines when a high-energy orbital loses one
electron into a lower-energy orbital, because the
simple drop-to-lower-level in energy only can
occur with emission of a photon, to conserve energy.
So, photons that are characteristic of an atom's
electron orbitals are seen in the aftermath of
unloosing energetic electrons.   There are many
ways the electrons can settle down to ground-state
atomic lowest-energy states, including X-ray emission
and infrared and ultraviolet, but also visible light.
The colors we see are characteristic not of the X-ray
input, but of the atomic or molecular orbitals
allowed by quantum mechanical principles.
A: In addition to the answer by Whit3rd :
The difference between materials that display phosphorescence and materials that display luminescence is that the mechanism of expelling the absorbed energy from the electron hits has a time delay in the first case.

There are two separate mechanisms that may produce phosphorescence, called triplet phosphorescence (or simply phosphorescence) and persistent phosphorescence (or persistent luminescence). Triplet phosphorescence occurs when an atom absorbs a high-energy photon, and the energy becomes locked in the spin multiplicity of the electrons, generally changing from a fluorescent "singlet state" to a slower emitting "triplet state". The slower timescales of the reemission are associated with "forbidden" energy state transitions in quantum mechanics. As these transitions occur relatively slowly in certain materials, absorbed radiation is reemitted at a lower intensity, ranging from a few microseconds to as much as one second after the excitation is removed.


On the other hand, persistent phosphorescence occurs when a high-energy photon is absorbed by an atom and its electron becomes trapped in a defect in the lattice of the crystalline or amorphous material. A defect such as a missing atom (vacancy defect) can trap an electron like a pitfall, storing that electron's energy until released by a random spike of thermal (vibrational) energy. Such a substance will then emit light of gradually decreasing intensity, ranging from a few seconds to up to several hours after the original excitation

italics mine for emphasis
