I was looking for a free program to play with frame-by-frame animation and my Bamboo tablet, and this seems to be it:
http://www.synfig.org/

Of interest, there's also commercial software such as Toon Boom (has several editions including "Harmony," which is used on broadcast cartoons), Anime Studio Pro, After Effects (it has "puppet tools" for cut-out animation) and Flash CC.

I was also seriously considering making my own animation suite with Java (with Swing, JPen and JOGL), but the effort necessary is just gargantuan. I'm part of the crowd that has the 'developer itch,' that we want to make our own tools, but in the end it's more efficient to use what's available, and Synfig Studio seems capable.
It would be an entertaining project nonetheless, making a 2D animation and motion graphics suite from scratch, and it wouldn't go to waste: choice is always welcome.

Well, that Synfig program has a canvas bug in the latest "stable" Windows build. Additionally, it crashed when I tried previewing a sequence.

So I'll have to keep looking for another solution.

To my surprise, the previous version of Anime Studio Pro (a decent animation package, but commercial) is sold at 10% the cost of the latest version.

As of this date, the latest version of A.S.P. is 10 and goes for this value:

http://anime.smithmicro.com/buy.html

But on Amazon you can find a listing of the previous version, which is still usable, such as this:

http://www.amazon.com/Smith-Micro-Software-Anime-Studio/dp/B00G7WP2R6/

In fact, if you buy that older version, you can then upgrade to the latest version with the upgrade cost: http://anime.smithmicro.com/upgrade.html

Then you will have the latest version of the software and the total cost involved in that will be less than buying the latest version by itself. That is odd.