Mode I is sRGB color, which is the standard color mode for web display.
Mode II is Adobe 1998, which is a wider gamut color mode. It is especially suited to inkjet printing, and comes closer to representing all of the colors that current inkjet printers can print.
Mode III is an "enhanced" sRGB mode which is like color mode I, but with increased color saturation. Think of it as "digital Velvia."
The sRGB color modes have a smaller color gamut than Adobe 1998, and may not be the best choice to take advantage of present and future color output devices. If you are using Photoshop or another image editor program that recognizes color profiles in general and Adobe 1998 in particular, you are best served by shooting all images in mode II to preserve the most color information in the image. If you are shooting for web display, one of the sRGB modes is the simplest route, as you don't have to convert color profiles prior to posting. Personally, I shoot everything in mode II and convert as needed to sRGB for web display.
You can get an idea of what the color modes look like on my D100 test page. The Adobe file will have to be downloaded and viewed in the appropriate image editor to appreciate the colors (which do not display properly in a web browser.)