* This is of course determined by how much visual information they contain — dimensions, number of frames, number of colours, etc.
Examples
Here's one I made many years ago (as seen here, posted by scigamerfan07):
And here's another I posted yesterday, illustrating the subtle differences in the unused Cranky's Video Game Heroes screen recently discovered by Kingizor:
Animated GIFs also help bring the DKC Atlas mapping project to life.
As you can see, they are handy for showing character animations, and for comparing images.
Limitations
Perhaps the main disadvantages of GIF files are the limited colour palette they offer (256 unique colours per frame), and the lack of alpha transparency, but when working with 16-bit video game graphics, these limitations are very rarely an issue.
Make your own
There are a bunch of different ways to create animated GIFs, but I would recommend using GIMP (Graphic Image Manipulation Program), as it is a powerful, free, multifunction program, and will be of more creative use than just assembling GIFs.
At some point I hope to provide a tutorial or two that are specifically tailored to sprite-based animations, but in the meantime, these links ought to help you get started:
Download GIMP
GIMP Simple Animation Tutorial
GIMP Advanced Animation Tutorial
Anyone wishing to create a more sprite-specific tutorial for GIMP, or for any other software, feel free to post it in this topic.
All questions, suggestions, etc. regarding animated GIF software and techniques is welcome here, too.
Sharing your GIFs
For those of you wishing to share GIFs you've created, we have the following topics:
DKC-Related Custom Animations (for customised DKC animations)
The Random Thoughts Topic (for all manner of nonsense, including non-DKC animations)
Note: This topic was created to be home to a discussion that had already started; the next four posts were made before this original post existed.