Roughly one third of diamonds available in the market fluoresce, like the fluorescent minerals you have seen in natural history museums, or the novelty shop toys under the black (UV) light. The effect is like a white shirt in a nightclub.
Fluorescence can be faint to very strong, and the most common fluorescent color is blue. As blue is the complimentary color to yellow, the most common tinted color in diamonds, blue fluorescence can make yellowish diamonds look white or colorless.
A GIA survey found that fluorescent diamonds were favored over non fluoro stones, especially in lower colors, but even in the higher colors (D, E and F) which are often discounted by the trade. Many years ago colorless fluorescent diamonds were highly prized and refferred to as "blue-white". But sales people used the term too loosely for any diamond with fluorescence; "blue-white" usage was outlawed by US trade practices laws.
One "for" argument for discounting fluorescent diamonds is because the GIA lab grading lights emit a small amount of ultra violet light; fluorescent diamonds might be assigned a better color grade. The ‘against’ argument, although the GIA Gem Trade Lab has not openly discussed the issue, is that UV light is almost always present in viewing environments, so why not grade color in realistic lighting?
But the most likely reason for fluoro diamond discounting is because of the sad fact that many jewelry salespeople are not able to explain complex phenomena like fluoro; a Fluoro (or any comments) written on a report makes the diamond harder to sell = worth less!
Some diamonds have extremely strong fluorescence and appear oily or cloudy. This is BAD. But the GIA study found them to be very rare; they were unable to find enough cloudy stones from the tens of thousands of samples they used.
Rarely diamonds fluoresce another color like yellow or orange. Do not buy them unless the diamond concerned is a fancy color of the same hue as the fluorescence (which will make it more intense). White diamonds with yellow or orange fluoro will appear to be a lower color when seen in light with a UV component.
When the UV light is turned off, fluorescence ceases instantly, but some stones continue to phosphoresce for a little while.