Aviary is an aptly-named suite of online image manipulation tools. It comprises several pieces: Phoenix, an image editor, Peacock, a tool for building filters and creating effects, Toucan, an app for building palettes, and still under development is Raven, a vector graphics tool.
Like Photoshop, Aviary's files support layers. That means that you can do all of the layer-y things that you can do with Photoshop files: hide and show layers, move them up- and down in relation to other layers (overlapping), and move them around the screen as well. What you don't have to do is pay a huge amount for the privilege. And getting started is easy: they've got lots of tutorial videos to help you see what these tools can do, and how to get started.
Aviary is a free service. You'll need a web browser, but not much else, to use it.