Somewhere between a nightly and a beta lies Firefox Aurora, Mozilla's first public browser under its new publishing schedule. Announced yesterday, Firefox Aurora for Windows (download), Mac (download), and Linux (download), is most analogous to the intent behind the Google Chrome developer's channel (download Windows | Mac | Linux): to provide a place where users who don't mind browser instability can test out new features, and contribute feedback to the developers.
Mozilla is now running all four of the browser builds that it mentioned in this blog post: Minefield, or the nightly test builds; the new Aurora; the beta build, which users who downloaded the Firefox 4 betas are still using despite there not being an official beta release at this time; and the Firefox final release, which is the stable version most Firefox users have.
Currently, Mozilla Firefox Aurora 5.0a2 doesn't appear to offer anything different from Minefield or the stable release, but that will change as Mozilla pursues a release schedule similar to Google Chrome's six-week release cycle.
If you do notice anything different between the current Aurora and stable builds, let me know in the comments below.