Load the image into GIMP2 then click Image - Mode - Indexed. For Colormap, select "Generate optimum palette", set "Maximum number of colors" to 16, and then click Convert. (Note: if your image uses less than 3 colors, GIMP may save it as a 2-bit color image, which will not be suitable for use in a download file.)
To specify the transparent color, click Colors - Map - Rearrange Colormap. A row of boxes containing each color used in your image will appear. Drag the box containing your background color into the left-most position, and then click OK. You don't need to specify a transparent color if you're creating a wallpaper.
To save your changes, click File - Save As. Set the Name as you wish. Hit Enter or click Save. When the "Save as BMP" dialog appears, make sure the "Run-Length Encoded" box is empty, and then click Save.