Originally Posted by Fetish Gimp

To reduce the filesize of a gif without changing its width and height you can try:

• Decreasing the amount of colors it uses. The less colors you use, the more it will affect the quality of the image.
• Decreasing the number of frames it uses. The less frames there are, the less fluid the animation.

Most image editing programs that can handle animated gifs (Photoshop, Gimp) will allow you to do those things.