http://www.humanbalance.net/gale/us/
We use Graphics Gale for drawing precise pixel art and making animated sprites.

http://www.gimp.org/
The GNU Image Manipulation Program is a powerful image editor useful for drawing sprites or textures for use on 3D models.

http://www.inkscape.org/
Inkscape is used for creating vector graphics, to design images which look smooth at any size. It's good for desktop publishing too - we use it for our leaflets and posters.

http://sketchup.google.com/
SketchUp is an incredibly easy to use 3D modelling program, good for beginners to have fun with. A cut-down version is free to use. However it is difficult to export the models for use in games.

http://www.blender.org/
A powerful, free modelling program. It is fairly difficult to start with but there is a comprehensive tutorial available.

http://chumbalum.swissquake.ch/
Another powerful modelling program, but this isn't free. It does export to a useful range of file formats for use in games.

http://developer.nvidia.com/object/fx_composer_home.html
Vertex and pixel shaders are small programs which run on a graphics card to produce next-generation graphical effects. FX composer is a powerful shader editor which shows what the effect will look like when applied in a game.

http://www.typhoonlabs.com
Shader designer is similar to FX Composer, but is designed for the OpenGL Shader Language (GLSL) .