Originally developed for Unix, GIMP has been ported to every major operating system in at least one form (and usually multiple ones), which makes it an easy blanket recommendation for anybody seeking all of the features of Photoshop with none of the expense.
I could end this article right here--except for the fact that GIMP is, rather notoriously, as slow as molasses. If you're a Linux user, or if you want free access to a range of tools that's similar to what Photoshop offers, then GIMP is still your best bet. However, Windows and Mac users just looking to remove the backgrounds from images or wanting to do some simple work in layers can turn to alternative free image editors that are faster.
Windows: You can find a great free image-editing package in Photoscape. In addition to an image editor, this software includes an image browser and a batch processor for quickly editing groups of photos at once. It's also a fast and stable little application. The problem is that it doesn't support any advanced image-editing features. If you need to color-correct an image, Photoscape is great--but its lack of layer support and its omission of advanced selection tools means that it falls short as a Photoshop replacement.
If you need an application with a little more power behind it, Paint.net is probably the answer. Although it still isn't as complete as Photoshop, Paint.net is much more powerful than Photoscape, and its support for layers and image filters should satisfy all but the most hard-core users. Paint.net is the fastest and most stable Photoshop replacement I tested for this article, too: During my limited time with Paint.net, it didn't crash once, and it loaded and manipulated images speedily.
Mac OS X: Though the freeware editor Seashore uses GIMP's image format, it was developed in Cocoa specifically for Mac OS X. That means Seashore doesn't require X11 to run its graphical user interface the way GIMP does for Mac OS X, and as a result it's significantly faster than GIMP.
Seashore does have a few disadvantages, however. First, it isn't as full featured as GIMP or Photoshop. Though it does provide layers and all the basic editing tools, it offers no support for plug-ins, and the filter selection is limited. On top of that, it isn't the most stable image editor; I've had Seashore crash on me more than once while I was in the middle of editing, so I suggest saving early and often.