Animation Creator HD (Full - $1.99 / Lite - Free)
I have been pretty impressed by this frame-by-frame animation design application. The interface is very user-friendly, and with time and attention you can create some really elegant animations. As you add a new frame, a 'ghost' of the previous frame is provided so you can manipulate the subsequent frame accordingly.
It is easy to import images and draw objects. Some downsides include a lack of an audio component and the fact that you can only export the video file to YouTube.
Animation Studio ($1.99 - iPhone compatible)
The interface is a bit clunky. It took me a while to figure out how to add new frames and test out the movie, but I think if someone spends enough time understanding it there is potential to create some pretty snappy movies. I especially like the ability to compose your own soundtrack using instruments and layers built into the program.
PhotoPuppet (Full - $3.99 / Lite - Free)
This is my third or fourth time mentioning this app since I started keeping track of my investigations on this website. The simple fact that you can animate objects without needing to do frame-by-frame design is a huge benefit, especially when thinking about having kids design animations.
One feature that I have mentioned to the developer that would be great is the ability to import sound clips and songs. You can record voice narration, but it would be a fantastic to be able to add sound effects and background music.
Photos taken from the iTunes store website page for each app.
Header photo by Robert S. Donovan