To see this underwater, you'll need a light with an "excitation filter" attached AND a "barrier filter" on your mask and camera system. What you'll generally see is a glow emitted from a variety of corals and animals, mostly in greens, blues, and reds. You may find two of the same type of corals—one which fluoresces, and one which doesn't. Dive your favorite spots in a different light and see what you've been missing!
Fluorescence is technically the photon emitted as an electron relaxes from its excited state to its ground state. In layman's terms, it's the glowing you are familiar with in glow sticks, jellyfish, and forensics. A variety of organisms and materials above and below water exhibit fluorescence—often with the introduction of a particular wavelength of light.
Fluorescence photography may be captured during the day by experimenting with using a very powerful, filtered light source and under-exposing the ambient light.