There are several ways to do it...
The bare minimum you will need is MS Paint -- or Mac equivalent -- and GIMP (which is free). Edit the sprites in MS Paint, copy each sprite over to GIMP as an individual layer, and export as a GIF. Make sure each layer is tagged with (replace) and that you set how long each frame should last (I do a uniform 50 milliseconds)
(If you go this route, I can't really help you with the actual "editing the sprites" step; Basically what you do is take pieces of other sprites you like, Frankenstien them together, and recolor as needed)
Alternatively, you can do what the Official DesktopPonies Team does and use this (requires Photoshop):
[link]Remember: the PrintScreen button is your friend, and anything saved in MS Paint should be saved as a PNG or BMP
only. Saving as GIF is ONLY done in GIMP, and
never save as JPEG
The Official DesktopPonies Team has their own DA page here:
[link]