Please note that converting an image with thousands or more colors to 256 colors is a destructive change that will result in decreased image quality. GIFs currently come in two flavors, GIF87a and GIF89a. The GIF87a format has fallen out of favor somewhat since it doesn't support transparent colors, interlacing, or animation. Both types support dithering. Both should be viewable in any browser that supports images.
Fewer colors equal smaller files.
GIFs have to store every color used in a table. The fewer colors stored, the smaller the resulting file. Interlacing a GIF will make it appear more quickly in a browser, but will make the file larger. A good rule of thumb is to interlace images larger than 10 or 15k, but not those that are smaller. You can dither a GIF and make it seem to contain more than 256 colors.