Championed by Linux users everywhere, GIMP (Gnu Image Manipulation Program) is an entirely free-to-use image editing program which is also available to run on Windows. Although never designed as a direct challenger to Adobe Photoshop it offers a rather similar user experience albeit with a much reduced, although ever-growing, set of features.
It won't catalogue and maintain your images - you can use the free Google Picasa application for that. Instead, it's a pure image editor with the emphasis on functionality rather than ease of use for the novice.
When you start it up you get a main window where you can open images or create them from scratch, plus a pair of windows: one a toolbox full of icons and the other containing brushes, channels, paths and so on. This layout will look familiar to those used to Adobe Photoshop, but will leave the novice high and dry. In general, the user interface offers the bare minimum of what's required to get the job done, and in some cases the user can be exposed to some rather cryptic-looking error messages if things don't go according to plan.
However, the large range of tools on offer makes it well worth putting a little effort into learning the program. There's full support for layers and masks, painting tools, sophisticated brush controls, animation and a handy visual undo history which lets you go back through previous versions of your image by selecting a preview thumbnail.
Being part of the GNU project, GIMP is subject to regular updates and enhancements as well as unofficial plugins and hacks with varying levels of success. Many functions not supported natively by the program, such as HDR images or GIF animation, can be handled by the use of plugins and add-ons - which you'll have to seek out for yourself.
Sometimes these plugins offer cutting-edge features before they're found in commercial packages. For example, Dr Paul Harrison's Resynthesizer plugin has enabled content-aware fills for nearly 10 years.
It does however lack a few important features such as adjustment layers and support for 16bit per channel images.