The Problem
Windows 256 colors are different than the 256 colors supported by Macintosh operating systems.
There are monitors capable of showing thousands of colors and ones capable of showing millions of colors and ones capable of only showing 256 colors. (The higher end monitors can be set to different color display settings.)

The Solution
Common Ground
The Web browsers support a 216 color system (a cross-platform, cross-browser standard)

How to Proceed
If you create images in this browser safe color system they will display the same on both operating systems and in both IE and Netscape browsers (regardless of the monitor settings).
If your color palate is not browser safe the colors in your images will be dithered.*
If you set your monitor to 256 colors as you create graphics for the Web you will see your images as the low-end monitors will display them.
Use browser safe colors for navigation graphics in gif format (will not be dithered) and full color jpg images for photos (will be dithered).

Exercise
Open a few images (gif and jpg) and view at 256 colors, thousands and millions of colors to see the differences.

Web Monkey article on browser safe colors

*If you use a palette that includes other colors (beyond the browser safe palette) then these colors will be dithered when viewed through a browser on 256 color monitors. Dithering is the process of juxtaposing pixels of two colors to create the illusion of a third color. (Your eye mixes these two colors and approximates the third, similar to the process that takes place when a Seurat pointillist painting viewed from afar.) This process creates the illusion of full color but sacrifices image detail.