There are some simple web based tools to do photo editing, you can find them in Chrome plugins or Firefox plugins. For more involved stuff the closest thing to Photoshop for open source is GIMP.
I've also used Paint.NET - it's only for Windows. It is not nearly as sophisticated but it does most of what I need it to do. GIMP is a much more involved tool and requires a certain amount of investment to learn how to use it.
If you are a professional designer its one of those "you are only as good as your tools" type situations. However if you need basic editing I have use Pic Monkey to to basic editing and re-sizing. I guess it depends on what you are using it for!
I have used PhotoPAINT from the CorelDRAW Graphics Suite in the past. It is not a 100% Photoshop replacement, but suffices for most common tasks and is pretty complete. For the price of Photoshop alone, you get a usable PhotoPAINT and the very decent CorelDRAW, which is more like Illustrator, with better support for multi-page documents, although it is not a full InDesign replacement.
I have mixed feelings about GIMP. Tried it a few times and it seems to be evolving really well, but it's Linux heritage and unconventional interface might put some people off. GimpShop was (still is?) an attempt to tweak the GIMP interface to be like Photoshop as much as possible.
And I heard very good comments indeed about PixelMator. Mac-only (and only recent OS is supported), but very cheap (around $24). To me, it seems mostly a replacement for a large set of the common Photoshop tasks for everyday users, without the depth of Photoshop features. It boots fast, works efficient and is totally Mac-oriented in the interface.
I use GIMP because it is free. I'm not sure I would consider it a viable alternative to photoshop. It does the job for what I use it for. (minor graphic manipulation on a occasional basis) I prefer photoshop, however, for the amount of work I do on it, it doesn't justify the price.