If you are searching for animation software for a tablet or phone, there is currently (2020) nothing better than stop-motion-studio available for both Apple and Android devices. Buy it if you can, to support developers, but the free one is very good.
The rest of this page was written in 2016 and needs updating :)
There are a number of packages available for both the Mac and PC; unfortunately none are perfect in every way - so you have to weigh up your requirements
Mac & PC
A lot of schools have turned to icananimate recently, mainly because the licensing is easier, but it also works well. There are some compromises in terms of functions and quality but this is an ideal package for young groups. No loop facility. Now available in a version for the iPhone and a pro version called proanimate which can loop, capture at very high quality (12mp apparently) and control a canon stills camera.
Hue have now designed animation software to go with their funky little web cameras. Our only bug with these cameras is that the auto exposure and auto white balance functioned terribly for animation, making your work flicker more than for any other camera. (this was early in 2010 so the problems might well have been ironed out)
On the Mac
Is a great piece of animation software.
Many packages have 'stop motion' in their title - don't let this mislead you, you can & should use them for drawn, cutout & pixilated projects too.
On the PC
Is a basic animation tool for the pc, its free and if you can get past the 'difficult-to-read' website it's a good tryout
Is another piece of software that a lot of schools like working with.
Most of these companies are now producing versions for the Apple App Store and/or Google play.