Software for making 2D art:
Photoshop or Gimp or both if you want, for your pixel needs.(Gimp is the free one)
Krita, for digital painting, the brushes are amazing.
Inkscape for vector art. Gimp, Krita and Phothoshop can do vector, however they aren't vector focused.
Any 2D animation software(Spine 2d) or Any 3D animation software (Blender); It's easy to make 2D assets with 3D software.
Good List. Just wanting to add one of my favorite drawing tools:
Manga Studio 5 / Clip Studio Paint: its a tool meant for Manga/Comic artists, but it packs some tools that make it interesting for other 2D works. Its brush stabilization makes it great when working with crappy digitizers (like the old wacom digitizers employed by some windows tablets, or the MS Surface N-Trig solutions)... smooths out all the wonkyness and imperfections at the cost of a little bit more lag (which depending on how fast your strokes are, you will not even notice).
Also quite nice for coloring, even though many things of PS are missing (most notably working in more than 8bit color depth).
Then there are a ton of nice speedup tools for comic artists which might not interest you much.
Other aspects of the tool are less stellar, the algorithm for scaling is quite crappy, resulting in blurry lines once scaled (compared to the still perfectly crisp lines you get in PS after scaling), the layer options are limited (cannot transform layers as a group for example), and so on.
Given we are talking about a tool that costs 30-70$ (depending on sales and stuff) vs. the still quite expensive PS (even with the sub), not bad at all though as far as value goes.