The principle behind fluorescence is that the fluorescent moiety contains electrons which can absorb a photon and briefly enter an excited state before either dispersing the energy non-radiatively or emitting it as a photon, but with a lower energy, i.e., at a longer wavelength (wavelength and energy are inversely proportional).
The difference in the excitation and emission wavelengths is called the Stokes shift, and the time that an excited electron takes to emit the photon is called a lifetime. The quantum yield is an indicator of the efficiency of the dye (it is the ratio of emitted photons per absorbed photon), and the extinction coefficient is the amount of light that can be absorbed by a fluorophore. Both the quantum yield and extinction coefficient are specific for each fluorophore and multiplied together calculates the brightness of the fluorescent molecule.