DoInk is a free online drawing and animation tool that runs right in your browser. You can treat it like Microsoft Paint and use it to do just a quick doodle, or take advantage of its layer cloning and vector-based designs to create relatively advanced animations.
I chose the latter, and put together a pretty slick looking animation in just a few minutes. Adding additional frames is simple and intuitive, and the app saves everything you're working on in the background (and in the cloud) so there are no local files to worry about. You can also hop between projects at any time, just like you would in a software app.
The app does a great job at cloning individual frames, and will show each one that came before it as a "ghost" that remains in the background while you work on the new one. If you notice one or more of them needs to be a little longer you can also begin cloning it several times right from the timeline that sits in the bottom of the screen. This lets you make minute adjustments like moving a character's mouth, or background imagery to create a realistic looking animation.
When done with any creation you can share it with the DoInk community, e-mail it to a friend, or embed it on a blog or social-networking profile. I've embedded one below that behaves like a YouTube video and won't start until you hit the play button. You can also publish anything you've made straight to YouTube, which is a nice touch.
Redball jumper by JoshLowensohn, made at DoInk.com