What are the top 10 applications to produce cartoon movies for children?

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here are several packages available that are used, from quite simple, to very advanced. Some are commercially available, others are proprietary in-house solutions.

 For example, Studio Ghibli's production team of Howl's Moving Castle (http://disney.go.com/disneypictures/cast& , http://www.xsibase.com/articles.php?deta& used Soft Image XSI (http://www.softimage.com/products/xsi/) to create the animation of several elements, like the moving castle. These days even in a 2d production a lot of 3d animation packages are used in the production pipeline.
 
Appleseed looks like it's a 2d cartoon movie, but uses cartoon. or cell shaders, to recreate a more or less hand drawn 2d character look. http://www.appleseedthemovie.com/
It is actually completely done in 3d software. South Park is animated using Maya (total overkill, if you ask me!). All the graphical elements are 2d, however, in this case.

 2D Packages used, are:
- Toonboom studio : http://www.toonboom.com/
- Anime Studio : http://www.e-frontier.com/article/articl&
- http://www.digicelinc.com/2Danimation_so&
 - http://www.bauhaussoftware.com/2D_Animat&
- http://www.plasticanimationpaper.dk/

 It all depends, of course, on the HOW of animation. Take Anime Studio, for an instance; this allows for a bones-like structure to be build up from graphical elements. This makes it easier to animate, but isn't really hand drawn animation.

 Digicel Flipbook and Plastic Animation Paper are a kind of digital flipbooks, albeit very advanced flipbooks. Don Bluth uses Flipbook, and loves it.

 Besides these, mainstream software like audio editors, music sequencers and video editors are often utilized in the entire process. (Adobe Premiere Pro, Avid, CubaseSX, Logic, etcetera).

 So, as you can see, the software in itself is not as important as your own input. Cut-style animation like South Park demands different software compared to completely hand-drawn animation. Some elements may be too hard or time-consuming to do traditionally, and a 3d animation app might prove useful in these situations.

 Anyways, if you're starting out, and do not want to dish out any dough, why not try these:

 Creatoon: for all-round animation, Creatoon is pretty much a full-fledged animation program & very powerful (view their samples!). Still quite easy to use::

 http://www.creatoon.com

 It's a free download now! So very friendly of the developers. They have a number of tutorials available.

 For AfterEffects like animation & videoFX, Jahshaka may be the (free!) answer:

 http://www.jahshaka.org/

 Not as simple to learn to use, because it's quite professional in its scope, but nonetheless very powerful. Also built-in is a good video editor.

 For a 3d animation solution:

 http://www.blender.org.

 VERY powerful 3d animation app, can be used to animate flat 2d graphical elements as well, off course. Another great thing about this app is a complete built-in video compositor.

 There are, obviously, also many free video editing solutions available. (Windows Movie Maker, Imovie, Avid free, Zs4 video editor: http://www.zs4.net/ to name but a few) and sound solutions (Audacity: http://audacity.sourceforge.net/).

 These should get anyone started in the wonderful world of 2d animation.