Basic Process
Objects absorbing UV use some of the energy, then may emit lower energy light. Visible light has lower energy than UV, so the light released may be any color.
Fluorescence
This process of absorption of one frequency and energy of light and almost immediate re-emission is termed fluorescence. This process occurs at the atomic level.
Don't All Objects Absorb Black Light?
In a major step forward, Niels Bohr described atoms as having electron "orbits." Different atoms absorb different specific amounts of energy. For some atoms, the energy of UV photons (particles) is just right.
What Causes Black Light Absorbers to Give Off Color?
Bohr proposed electrons can also switch to stable, lower-energy orbits by emitting photons. Some emitted photons may be seen as colors.
Example
You shine UV light on three objects. Two absorb it, one doesn't. The one that doesn't can't be seen under these conditions. For the other two, one gives off visible light. The last one emits photons possessing too little energy to be visible light. Therefore, you see one object.
Conclusion
Quantum physics determines which objects are seen under black light.