Free software:
Synfig Studio for 2D animation; free alternative to Adobe Flash and Toon Boom.
http://synfig.org
Krita for concept art and illustration; free alternative to Corel Painter.
http://krita.org
MyPaint for drawing and free-form painting; free alternative to Paint Tool SAI.
http://mypaint.org
Time-lapse videos on YouTube:
Aqua Dream (Krita)
http://youtube.com/watch?v=dsDgtIWXLXE
Portrait (MyPaint)
http://youtube.com/watch?v=MAQv3eYgXzo
Watched Wave (MyPaint)
http://youtube.com/watch?v=GmxPdUXMWBA
Blender for 3D animation; free alternative to Maya, 3ds Max, and After Effects.
http://blender.org
Sintel (Blender)
http://www.youtube.com/watch?v=eRsGyueVLvQ
General Sherman: Is this for real? (Blender)
http://www.youtube.com/watch?v=GcYdgudn83M
All listed free products are 100% free with full functionality, no adware, no spyware, and no viruses, and you're allowed to use them in any production without having to pay for anything.