Many diamonds exhibit fluorescence under ultraviolet light, this is similar to white clothing glowing under a night club black light. We use the property of fluorescence to prove a diamond is "real", however the reverse is not true, as not all diamonds exhibit fluorescence.
The picture on the right is a three stone diamond ring that has a ultraviolet light pointed at all three stones. The stone on the left is exhibiting blue florescence.
The most common fluorescence color is blue however they can also exhibit orange, red, white and yellow. Natural sunlight contains ultraviolet light and therefore can clause diamonds to fluoresce.
You will not see fluorescence under a jewelers shop lights! We use ultraviolet lights (available as button and small flashlights) to look for fluorescence. Fluorescence can cause a diamond to appear milky or oily under some light conditions.
Diamond fluorescence can impact the value of the diamond, but does not impact as much as cut, clarity or color. Fluorescence should be listed on the diamonds "cert" or grading report. All colors except for blue should be avoided; blue florescence can be tolerated at a "slight" level and under. Blue florescence may actually improve the face up white appearance in diamonds with a color grading of H and below.