Absolutely essential

A computer with decent 3D graphics capability
The vast majority of recent computers meet this criterion. A recent version of Google Chrome reportedly can run WebGL on the CPU, so it may be possible to do very basic work on a computer without a dedicated graphics chip, but you will not get the best possible results this way.
A browser that supports HTML5 and WebGL
Chrome is highly recommended. The next best choice would be Firefox. You might also have luck with Opera or Safari. Until recently, Internet Explorer was not working, but the latest version appears to work, at least partially.

Necessary for model creation

A 3D modeling tool
Blender (currently version 2.78a) is recommended.
A 2D painting tool
Gimp is more than adequate for most of us. The majority of professionals seem to use Photoshop instead. You may wish to download and install the 32 bit normalmap plugin as well.

Nice to have

A good programming editor
Notepad++ is my favorite for Windows. jEdit is a good multiplatform choice.
Image processing utilities
Image Magick is very handy when you need to quickly translate a set of textures from one format to another, for example.
Sound editor
Audacity is recommended if you want to edit your sound files.
An automatic texture generator
ShaderMap costs money but works well and has a free command line version (ShaderMap CL), but is Windows only.
Three's source code
The source code (we'll use r82) is the ultimate answer to all questions about Three functionality.