The cartoon animation software listed in this page are vector based.
Unlike the Classic Animation Software that are designed for drawing animation frame by frame, a vector based software lets you use the computer to do some of the hard work.
Flash animation is a popular example of this genre. It is not, however, a dedicated animation software, but a web-authoring tool. Happily, the new FlashCC software is much more "animation oriented", with a motion editing pane similar to the one in After Effects.
ToonBoom is an entire family of animation softwares for all levels, from kids to pros.
It's up to you to know what you need.
Try the trial version of Flash, or free lite versions in Toonboom, so you can see what works best for you.
I've also listed below a number of completely free cartoon animation software.
As with all free software, they vary in complexity and the features that they provide.
If you've never worked with symbols and "tweens" before, you might need a little practice to get the idea behind this technique.
Do browse the Flash cut out animation tutorials in this site to get an impression of how it's done.
It might be better to get the hang of it with a free software, before investing a couple of hundred dollars in a big professional program.
Be careful - it's addicting :-)
I started out as a classic animator, and didn't even want to look at cut outs, but once I started I got hooked on the marvelous things you can do with symbols.
Here's a little peak into the Flash spirit... speaking from the GUTS, I'm sure you know this one, but I thought I'd put it here anyway:
An amazing family of programs for making animation, for animators of all levels.
Sooo many people have discovered this software and go crazy with it.
Just search YouTube for "stickfigure animation" and see for yourself!