External Tutorials
On this page, there are links to several tutorials covering all manner of artwork, including sprite art. No guarantees on how long the links will last, but then, the beauty of using a wiki is that anyone can submit corrections to them. Hopefully, this should provide a very good base of knowledge to work from. None of these can substitute for practice, rather these exist to guide your practice, and help it to improve your skill rapidly, rather than having you flog away at simple cartoon forms for years and not learn anything.
As stated in some of Jetrel's other tutorials here, do not jump into the sprite tutorials and ignore the works on general or figure drawing. If you intend to draw sprites well, the ability to draw and visualize the subject in a general sense is an absolute prerequisite. Without exception, the best sprite artists are such because they have good basic art skills; understanding of shading, light, proportion, edging, foreshortening, etc.
You can't fake those - even if you never practice anything but spritework, the tutorials for "higher art" are very applicable, and will improve your spriting skill considerably.
General Art Instruction
Andrew Loomis Books
Andrew Loomis is a master of his field. Read his books! They are classics, and are of pristine educational calibre. If you have any interest in drawing human beings though, they will help you more than the rest of the tutorials on this page, combined.
They unfortunately have been out of print for a long time now, and will not enter the public domain any time soon. Copies are highly priced and hard to find in most cases. On the other hand, due to demand, pirated copies are widely available online — it’s just a fact, we don't endorse piracy.
You can start with Fun with a Pencil and Figure Drawing for all it's Worth.
Other Resources
For anatomy George B. Bridgman actually taught Andrew Loomis and is was well regarded for his contributions to art education, his books are still in print and pretty cheap, if you can pick up the smaller individual books I highly recommend doing so, "Bridgman's Complete Guide to Drawing from Life" is a large collection comprising of most of the same drawings and text but due to the changed order and layout is not nearly as good as the originals, luckily some of them have recently entered the public domain and are available from the Internet Archive.
"Constructive Anatomy" seems a good place to start, "The Human Machine" is much more useful as a reference.
http://www.archive.org/details/constructiveanat00briduoft
http://www.archive.org/details/humanmachinethea009564mbp
Condensed human figure tutorial by DeviantArt member coelasquid.
http://coelasquid.deviantart.com/art/How-to-draw-all-sorts-of-crap-104342407
The Kodak company made the following, very comprehensive introduction to composition. These rules apply to pictures as well, and will help you when laying out the subjects in a picture.
http://asp.photo.free.fr/Composition/photoProgramCompMainClass.shtml
The following site has a very high-level and overbearing tutorial on color theory. It's quite useful to experts, but will probably confuse a beginner.
http://handprint.com/HP/WCL/wcolor.html
Itchy Animation has, amongst other things, a tutorial on the rarely-covered subject of the behaviour of light and its effects on color (rarely covered from an artists' perspective, that is).
http://www.itchy-animation.co.uk/
http://www.itchy-animation.co.uk/tutorials/light01.htm
GFXartist.com has several tutorials from different artists, including the painter Craig Mullins, who has, for nearly a decade, been the favorite artist of wesnoth's art slave "Jetrel," and has a good chance of keeping that position for the rest of either fellow's time on this earth. (J- I can't say enough good things about that guy; I would honestly rank him among the greats, like Rembrandt, or Picasso - he's *that* good. The tutorials posted on that site do him little justice - for a better sample, his personal site is http://www.goodbrush.com/ , which has over a thousand of his works.)
http://www.gfxartist.com/features/tutorials
The excellent pencil artist Mike Sibley has a series of tutorials on techniques he uses on his site. Though some are a bit specific to drawing with graphite, his discussion of negative drawing is very useful to any artist.
http://www.sibleyfineart.com/index.htm?tipsndx.htm
The following website, titled "portrait-artist.org", offers quite a few good tutorials on portrait drawing. Though it might not live up to the 'tall order' implied by its name, it's fairly comprehensive:
http://www.portrait-artist.org/
An excellent skin tutorial by DeviantArt member acidlullaby.
http://acidlullaby.deviantart.com/art/Skin-Tutorial-90966884
This website already features a couple of great painting tutorials and has potential to grow.
Tutorials on using Computer Graphics Software
Good-tutorials.com has over 8000 various tutorials covering different aspects of the popular program, "Adobe® Photoshop®."
http://www.good-tutorials.com/
The official collection of GIMP tutorials on www.gimp.org. Includes tutorials for users of all skill levels, from beginner through expert.
http://www.gimp.org/tutorials/
Sprite Art
The medium used to create sprite art is known as 'pixel art'. If you do not understand what pixel art is, or if you have only a general understanding of what it is, I suggest reading The Pixel Art Tutorial by Cure. It is comprised of a series of sequential forum posts on the www.pixeljoint.com forum and gives an in-depth overview of pixel art, explaining in detail what pixel art is and what it is not. Although aimed primarily at those who are new to pixel art, the tutorial can also be helpful to more advanced pixel artists.
An excellent pixel art tutorial that covers a lot of fundamental spriting and pixel art concepts, from sketching and selout to palettes and antialiasing. HIGHLY recommended for pixel artists of all skill levels.
http://purloux.com/artwork/tutorials/rundown/
A very worthwhile mini-tutorial by veteran spriter and pixel artist (and Wesnoth contributor) Neoriceisgood, in which he explains his spriting workflow step-by-step, from the initial sketch to the finished product. (Note: the tutorial is in the form of a 1208x1660 .png file.)
How_to_make_sprite_by_Neoriceisgood.png
Zoggles has a wealth of tutorials made by many different people, some good, some bad.
http://www.zoggles.co.uk/asp/tutorials.asp?show=index
A Russian site featuring several tutorials - you may need to click on the "English" link to view the site in a language you can understand, unless you have the privilege of knowing Russian.
http://www.gas13.ru/tutorials/
A fellow named mark posted several tutorials at the following site. Most of them are fairly decent:
http://www.natomic.com/hosted/marks/mpat/
The following is a forum post on some spriting forum containing a list of links to different pixel tutorials. Some are of banal quality, your mileage may vary.
http://pixeljoint.com/forum/forum_posts.asp?TID=706&PN=1
Animation
Carin Perron has posted an excellent tutorial on the basic principles of animation.
http://www.writer2001.com/animprin.htm
A very worthwhile tutorial on creating walking animations, covers essential details such as rotation, movement arcs, etc.
http://www.angryanimator.com/word/2010/11/26/tutorial-2-walk-cycle/
Another walking animation tutorial:
http://www.dungeoncraftgame.com/blog/?p=39#more-39