Before I start, a quick note on graphics image types. There are two major types of graphic data: raster and vector.
Raster is a map of bits (a "bitmap") where each pixel (dot) is one piece of information, stored digitally as a colour. Raster images are great for storing "noisey" data such as photographs and other images, but scale poorly (they get blocky when you scale them higher).
Common raster formats include JPG, PNG, GIF, TIF, BMP, etc.
Vector is graphic information that is stored mathematically in information such as lines, shapes, colour fill and gradiants. Vector information is preferred in the print industry, as it allows images to scale without losing any quality, and makes prints come out clear and precise without blockiness or artifacting. Vector information works well at storing simple "illustration" style graphics where colours are uniform and/or in even gradients.
Common vector formats include PDF, EPS, PostScrpt, AI (Adobe Illustrator) and CAD formats like DWG, DXF, etc.
GIMP (GNU Image Manipulation Program)
http://www.gimp.org Raster editor.
The grand daddy of non-free image editing is of course Photoshop. It's so popular it's now a verb! ("Hey, did you see that picture where that guy photoshopped a bunny to put a pancake on it's head?").
GIMP is an alternative package that comes pretty close to providing the same functionality. Available on Windows, Mac and Linux it caters for the needs of most folks easily. For a low-end user like me, I can resize images, save them as different file formats, happily import from formats like PDF or Photoshop PSD, colour correct, change colour channels and other filters, do red-eye reduction, and all the usual tricky tools like layering, clone stamping, cropping and warping, etc, etc.
GIMP's biggest downfall is that it only supports RGBA (Red Green Blue and optionally Alpha). This is fine for people who only use images for output to TV/monitor (eg: websites, movies, etc). For print people, the CMYK (Cyan Magenta Yellow blacK) colour space is missing. GIMP devs have been saying it will be added for a while now, but so far no dice.
GIMP can still happily print to a colour printer of course. I print all of my arcade artwork via GIMP and am happy with it. For professionals who need proper industry-quality colour correction, it unfortunately falls short.
All in all it's a great package and for 99% of the population who only use 10% of the features anyway, a perfect substitute for Photoshop.
http://www.inkscape.org Vector editor.
Similar in style to Adobe Illustrator or Xara Extreme, this is a simple vector editor that's perfect for folks wanting to output clean SVG or PDF information for web use or print.
It has some very handy tracing software built in to convert Raster to Vector and enabled you to scale to your heart's content. I wrote a thread with some examples here: http://www.aussiearcade.com.au/showthread.php?t=1858
Allows drawing of simple line objects, or complex gradients and transparencies. I use Inkscape frequently for drawing my own arcade sideart and marquees, and the results are quite nice.
Inkscape imports Adobe Illustrator files, and freely outputs to all filetypes including PDF, EPS, SVG, etc.
http://www.scribus.net/ Page Layout
Akin to Microsoft Publisher, Scribus is a free page layout and desktop publishing application. Handy for all sorts of weird documents, like resturant menus, kids school assignments, newsletters, multi-coloumned text, etc, etc. Outputs to PDF which makes mass printing easy.
Open Office Draw
Already mentioned above in the "office" section, so I won't go into detail here again. These cover the basic needs of most non-graphic-artist users who need simple image editing and drawing functions to add to their office documents. Has recently added support for extended colours systems, including 16bpp RGB as well as CMYK.
http://www.blender.org/ 3D content creation, editing and rendering
Similar to 3D Studio Max, Maya and other 3D systems, this allows creation, rendering and animation of anything, limited only by the user's skill. It's been used in some big-budget movies in hollywood, and is picking up steam in the 3D community.
Creation and rendering tools like this cost literally thousands of dollars per single license. Blender is free (as in freedom and as in free beer), and just as powerful.
As usual, Windows, Mac and Linux all supported.
http://www.qcad.org/qcad.html 2D Computer Aided Drafting/Drawing
Simple 2D CAD package similar in functionality to the early days of AutoCAD (before they went all 3D). I use QCad entirely for all of my arcade machine design and builds, arcade joystick designs, laser cutting, etc, etc.
It works only in DXF format (an older format supported by most CAD programs on the market), and is a suitable format for most engineering firms and laser cutters if you want to provide them with a digital file to work from.
Most Linux distros will include QCad's "community" version (comes with no commercial support, but is identical to the commercial version). Source is available and will compile cleanly on Mac (I think it's also in FINK). For Windows, you'll need to either compile it up yourself in MinGW, or buy a pre-compiled version. I've been meaning to get off my arse and build a windows version for some friends, but never got around to it. Most of them are happy enough to dual-boot Linux to use CAD anyway.