Fuoresence is related to the atomic structure of the atoms in the mineral. All elements will absorb light. The frequency of light absorbed is dictated by their specific electronic configuration in the atomic base state. For fluorescence, a photon of energy is absorbed by an atom by allowing one of the outer shell electron to be boosted from its natural excitiation state to a higher one. This transition is short lived with the duration of the higher state being dependant on the energy of the photon absorbed. When the electron falls back to the base state, the energy is released but the ewave lenght is now changed. The new wavelenght of light emitted is dependant on the change of state that the electron performs in the fall.

Joe-Speedy's answer gives a good explanation of what causes fluorescence, but he does not explain why some mineral won't fluoresce in long-wave UV light, but will in short-wave UV light. The reason for this is because of the nature of light and the energy transitions involved in fluorescence. First of all, let's look at the nature of light. Light has both wave-like and particle-like properties. In many ways, light behaves like the waves that you see in a body of water. (It spreads out from its origin and can be bent just like waves.) However, the energy of the light seems to be contained in discrete packets that we call photons. The amount of energy that each photon contains is determined by the wavelength of the light. The shorter the wavelength, the more energetic the photon. Now for energy transitions, iIn order to make the transitions that Joe-Speedy described, the electrons have to absorb specific amounts of energy. Electrons can't absorb random amounts of energy. The must absorb photon of just the right amount of energy to equal the difference between the original energy level and the new energy level. If the photon does not have just the right amount of energy, then it will just bounce off of the electron rather than be absorbed. So, let's put it all together. In order for fluorescence to occur, the electrons must be boosted to a sufficiently high energy level. Just how high that energy level is depends on the atoms and/or compounds involved. In the case of minerals that fluoresce in short-wave UV but not long-wave UV, the long-wave UV photons lack the energy to boost the electrons high enough to cause fluoresce whereas the short-wave UV photon do have enough energy.