Well first you're going to want to download
unFREEz which is basically a program where you drag all of the .gif frames to and it makes an animated .gif
You can use programs like MSPaint, Paint.NET, etc. to make .gif frames to put into unFREEz. Just save the images as .gif
After you do that you can set the delay for each frame in the unFREEz program.
then just save the animation to wherever.
If you're going to make a .gif animation for your avatar then you'll have to make sure each frame is at the most 75x75 pixels otherwise it will just stay as the first frame and will not be animated on the forums.
I would suggest using Paint.NET for the image editing, but that's just me.
If you need a picture tutorial I'll be glad to help :l
Sorry if this tutorial/whatever is a bit sloppy, I'm pretty tired right now.