For explainer videos, simple animated gifs etc. that you put on your website for example. The most recent example is the GitHub LFS animation: https://github.com/blog/1986-announcing-git-large-file-storage-lfs
I've found a few options but none seem to make quite as polished animations as I've seen around the web.
https://www.youtube.com/watch?v=eU6BwCPv3Ak
http://www.dummies.com/how-to/content/how-to-create-your-fir...
If you're comfortable doing it in code there's also JavaScript/HTML like this library which came from / uses Flash methods but I'm not sure how you go from this to gif:
https://greensock.com/get-started-js