In general, graphic designers are lucky folks. They get the fast computers and the big monitors displaying billions of colors. Because of this, we tend to get sloppy and assume that everyone can see the colors that we can.
This obviously isn't the case... the most popular web-browsing computer today is a PC-compatible, with Windows, running Netscape, and displaying 256 colors.
To optimize your graphics for display on such a system, you need to use a specific color palette of 216 colors. You only need to worry about these "safe colors" when using the GIF format, and only when you have contiguous areas of similar coloring (like on a button). Why are there 216 safe colors and not 256? Colors are represented as having different values of Red, Green, and Blue. For each of these components, there are six safe values, which gives 6x6x6=216 possible colors. The other 40 differ depending on the specific computer, so there are only 216 colors that are guaranteed to display right on a 256 color monitor.
Following is the palette of the 216 safe colors. Any colors not in this set will probably be dithered on normal displays.