A quick background: Google recently bought previously mentioned EtherPad with goals of integrating it into Wave, then closed the service. Then, after some backlash, they open-sourced the entire app. TypeWith.me took that code and put it on a new server that, unlike EtherPad, won't be closing down any time soon.
Getting started with this free app is as simple as clicking the New Document button on the site's home page. You're instantly dropped into an edit-ready page where you can start typing right away, or import an HTML, plain text, RTF, or Word document.
Each online document has its own unique URL you can share with others for real-time co-editing, and there's a small window for chatting on your edits. TypeWith.me helpfully assigns different colors to each author so you can easily track who's typed what, and the app instantly synchronizes as you type so changes appear in real time. When you're done, export your work to one of six file formats, including Word, PDF, and HTML.
Google Wave users will see a lot of similarities between it and TypeWith.me, but this simple app has an easier learning curve.