Black lights Emit safe UVA light and some violet light. These on the more energetic section of the electromagnetic spectrum. Some chemicals will absorb the energy from the UVA light and almost immediately re-emit the light. Thanks to the laws of Thermodynamics some energy is lost in this transaction and the light produced by the chemicals are in the visible spectrum.
Black light tattoo inks contain phosphors. A phosphor is any substance that emits visible light in response to some sort of radiation. A phosphor converts the energy in the UV radiation from a black light into visible light.