Big Note: It is possible to make graphics only in a program like MSPaint or Appleworks. There are, however, two major and vital things lacking in those programs - first, these programs cannot make images with transparent pixels, and second, they may not be able to save in the PNG format used by Wesnoth. A program capable of those will have to be applied to images made with MSPaint when they are finished.
Free Image Editors
The following are free image editing programs which can be used to create graphics for wesnoth. These programs are Open-Source Software, like wesnoth, meaning they are free to use, and that you are free to look at the sourcecode.
For windows and linux, the program known as "The Gimp"; the "Gnu Image Manipulation Program" is recommended.
For the macintosh, this port of the the Gimp to a cocoa-based gui is recommended.
For the macintosh, the OpenSword Group's "Pixen" is a great tool for traditional SNES style sprite art - unlike most other editors, it has been designed for that specific task. This program was written in cocoa, the best and truly native mac gui, and its interface creams any of the other programs listed here - version 3 will even include built-in support for making animations.
Proprietary Image Editors
For the macintosh, Lemkesoft's GraphicConverter is an excellent program for preparing and compressing png images for the game, and may also be useful for the creation of images. The shareware fee is $30, although large parts of the program are fully functional for free.
For windows and the macintosh, the famous Adobe Photoshop and Photoshop Elements are available for the preposterous prices of ~$700 and ~$100, respectively (the prices vary depending on where you get them). If you can acquire the use of these programs through a business or academic situation, they are extremely powerful, and are more than capable of some very advanced sprite techniques which elude simple bitmap programs - the price, however, is likely too much of a barrier to entry for most contributors. Eleazar recommends resaving all final PNGs made in Photoshop in it's companion program "Image Ready." These resaved files will be many times smaller.