LICEcap is a simple but effective Windows and Mac screen capture tool which records your desktop activity, and saves it as an animated GIF.
The program is very easy to use. Launch the program and there's no clutter, no menus, no complex settings: just a window which you can resize and drag to whatever area you'd like to capture.
Once that's done, clicking "Record" gives you a few useful options. So you might enter a title, which will be displayed at the beginning of the animation. LICEcap can highlight mouse clicks, handy for tutorials. And you can specify how many times the animation should repeat (or just leave it to run forever).
Enter a file name, click Save, and LICEcap immediately begins recording the area you've specified. And so you'll need to launch a program, open a dialog, browse some menus, or whatever else you're intending to do. Once you've finished, click Stop, and the animated GIF will be saved, ready for use.
There are plenty of issues here. You can't capture the full desktop, for example. Animations of any size or length will produce very large files. And GIFs only support 256 colours, so they're not going to look great.
Saving the animation to a regular image file does mean you'll be able to use and play it just about anywhere, though, and LICEcap's simplicity is another major plus. It's definitely worth a look.