|| In order for a GIF to be displayed with a transparent background it must be saved in GIF89a format as opposed to GIF87a. The easiest way to do this is with a program like Paint Shop Pro or LView Pro. However there is also a free program called giftrans.
First you must make sure your background is all the same exact color. Furthermore it must be a color that does not appear anywhere else in the image. In the example on the left I had to change the background to yellow because white appeared elsewhere in the image and those pixels would have turned transparent as well. Be aware that there are many indistinguishable shades of white as well. If your image has such a background, you can adjust the match tolerance when you replace the background color. Of course if you are creating the image from scratch you can set the backgound color right from the start. Don't forget these are GIF's and as such are limited to 256 colors.
The next step is to save it in GIF89a format. In general there will be a dialog box where you specify the color to be rendered transparent. You can also choose interlaced or not.
Well, that's it in a nutshell. The specifics will vary depending on the program you use. Consult the help section of your program for details.
Both Paint Shop Pro and LView Pro have a free evaluation period. giftrans.exe is a free command line program that will convert files from GIF87a to GIF89a format.