The gif file format can only have one transparent color so some concessions have to be made when saving your work in gif format.
The following images were created in Photoshop and saved in gif format with transparency turned on. A variety of colors and opacity levels were used. The only difference between the two of them is the background. One was save with an opaque background and the other with a transparent background.

The first image is a gif with no transparency at all. It appears to support multiple transparent colors because the all layers were blended before the image was produced. However, the image was rendered against a solid white background that cannot be removed.
The second gif has a transparent background but in order to get this, some serious sacrifices had to be made. A decision had to be made for all the colors that were partially transparent over the background. The colors had to become either completely transparent or opaque. As you can see the yellow became transparent and the blue became opaque.
The take away: Leave the background in and you shouldn't have to worry about transparency. However, your background will be visible if it is not the same color as your web page.