Krita is a free app for digital painting but also supports 2D animation. It is made by artists that want to see affordable art tools for everyone.
Krita
Live2D
Live2D has a bunch of software that allows 2D artists to create animations from their artwork. Available in both FREE and PRO versions.
MakeHuman (FREE)
MakeHuman is a free open-source software that allows you to create and pose 3D human characters. Useful for pose reference.
OpenToonz (FREE)
OpenToonz is a free open-source software for producing 2D animation and is known for being used by Studio Ghibli
Source Filmmaker (FREE)
Source Filmmaker is a free movie-making tool built and used by Valve to make movies inside the Source game engine. It is a full package and can be used to make cinematics using Valve models and environments.
Kinovea (FREE)
Kinovea is video player that allows you to slow down, study, draw and comment. Primarily made for sports enthusiasts but useful for animators too.
Animation Paper (FREE)
Animation Paper is a popular free software for hand-drawn animation created by Niels Krogh Mortensen
Daz 3D (FREE)
Daz 3D – A 3D morphing, posing, animation and rendering suite, normally retailing at $249 but is now free to download
Pencil 2D (FREE)
Pencil 2D – An animation software for vector and bitmap graphics. It is free, multi-platform and open source
Animator's Resource Kit 