Adobe recently released into public beta (ready to go, though, for all intents and purposes) their online Photoshop Express application. A further distillation of Photoshop Elements accessible via a web browser with
Adobe's Air technology installed Flash support (thanks to reader corrections - this is an Adobe Flex application, not Air), Express is no GIMP, but it's free, available across platforms, and serves the needs of the MySpace set very well.
Sure, you can make your head really big, but all of the basics (red eye reduction, retouching, cropping, etc.) are all available for the cost of a registration (meaning free). While Adobe is looking to monetize Express through some sort of premium service later on, it appears that Express will remain free in this form.
For the purpose of most teachers and students posting content to the web or editing pictures for presentations, reports, etc., Express provides plenty of features in an easy-to-use package with 2GB of web storage/sharing for your photos.
One caveat: 64-bit Linux support for Flash/Flex is still a bit of a pain, although this could certainly be considered a cross-platform application.
no Linux support for Adobe Air yet, so this is basically a Windows/Mac party. (thanks again, Matthias, for the clarification). While it would seem largely irrelevant on the Mac platform due to iPhoto, it does allow users to move between machines/platforms and keep their photos and editing work in the cloud (for free).