Raster Based Art Software
Photopea - A free, web based tool that works very much like Photoshop.
Autodesk Sketchbook - Free tool for quick sketches or fully finished artwork!
FireAlpaca - Free digital painting software.
Clip Studio Paint - Popular among artists, CSP has a free trial and costs $49.99, with frequent sales.
Adobe Photoshop- Requires Adobe Creative Cloud subscription. Photoshop may be all you ever need. You can paint, edit images, draw sprites and much more.
Gimp - Free alternative to Photoshop.
Krita - Free sketching and painting program for concept art, texture or matte painting, illustrations and comics.
Corel Painter - If you are hardcore into painting, Painter will give you some features Photoshop lacks ($399).
Art Rage - A good $25 alternative for making fine art pieces.
GraphicsGale - Free Pixel Art software.
Pyxel Edit - $9 pixel art editor designed to make it fun and easy to make tilesets, levels and animations.
Aseprite - $15 animated sprite editor and pixel tool.
Vector Based Art Software
Adobe Illustrator - $599, Free 30 Day Trial. ]Illustrator is preferred if you are drawing in vector format. You can import vector art originally generated in Flash, which is handy if you want to put something on a t-shirt.
Inkscape - Free and open source software vector graphics editor. Its goal is to implement full support for the Scalable Vector Graphics 1.1 standard.
Sumo Paint - Free browser based drawing program.
3D Art Software
Blender - Free, used in The Secret of Kells.
Maya - $3495 (Free Trial), used in Ice Age and James Cameron's Avatar.
3D Studio Max - $3495 (Free Trial), used in I-Robot and X-Men.
Modo - $995, used in District 9.
ZBrush - $595 (Free Trial), for modeling only, no animation. Good for intuitively painting on textures and color.
Mudbox - $745 (Free Trial), for modeling only, no animation. Good for intuitively painting on textures and color.
Swift3D - $249, allows you to create and/or animate 3D objects and export for Flash. We recommend creating your initial models with alternative software.
Other Art Software
Apophysis - A free 3-D fractal program that can also be used for animating dynamic fractals.
Also see our Art Tutorials!
Special thanks to @Art101 for organizing a number of these software suggestions!