An open-source alternative to Photoshop that debuted on Unix-based platforms, GIMP stands for GNU Image Manipulation Program. Today it's available in versions for Linux, Windows, and Mac.
GIMP offers a wide toolset, similar to Photoshop in many ways, and is a great alternative if you're looking for a no-cost image editor. The interface differs somewhat from Photoshop, but a version of GIMP is available that mimics Adobe's look and feel, making it easier to migrate over if you're ditching Photoshop.
The full suite of tools is available here - everything you're accustomed to is within easy reach, including painting tools, colour correction, cloning, selection, and enhancement.
The team that oversees development has worked hard to ensure compatibility too, so you'll be able to work with all the popular file formats without any trouble at all. You'll also find a very capable file manager built in, along similar lines to Adobe's Bridge.
Overall, this is a great option whether you've either got a limited budget, or want to move away from Photoshop for other reasons.
Paint.net is a Windows-based alternative to the Paint editor that Microsoft shipped with versions of Windows. Don't let that put you off, though, as it's a surprisingly capable and useful tool, available completely free of charge. The software started out life as a Microsoft-sponsored undergraduate project, and has become an open source project maintained by some of the alumni.
The focus is on ease of use, and there's a definite tendancy towards photo editing rather than artistic creation. That said, there are a range of special effects available, allowing you to easily create fake perspective, blend and push pixels around the canvas, tile and repeat selections, and so on.
A good range of selection tools, support for layers, and adjustments such as curves and brightness/contrast mean that Paint.net is a great alternative to Photoshop for photo editing, especially if you can do without some of the more recent additions to Photoshop's toolset.
Pixlr claims to be "the most popular online photo editor in the world", which may have something to do with the fact that it's free.
But it also boasts more than 600 effects, overlays, and borders and lets you do all the main things you'd expect from a photo editor, from cropping and re-sizing to removing red-eye and whitening teeth.
And if you're used to using Photoshop, then you'll find Pixlr's user interface easy to pick up quickly, as it's very similar. This free app is available in both iOS and Android varieties.