Private Browsing works similarly to Google Chrome's Incognito, or Safari's setting of the same name. Go into Tools on the Menubar and click Private Browsing, and all your movements on the Internet will not be recorded. Firefox's version of the feature saves all your tabs and closes the browsing session, re-opening a new and empty browser window. Unlike Chrome, Firefox doesn't sport a clever little icon indicating that you're browsing on the sly. The program title bar does state that you're using Private Browsing, but there's no blatant icon.
Firefox has also posted the about:config setting needed to set Private Browsing as your default browsing configuration. Once you're in the about:config, type in browser.privatebrowsing.autostart and double-click on the False setting. This will change it to True, and when you restart your browser you will automatically be in Private Browsing mode.
What's interesting about this is that it removes the indicator from the Title Bar and grays out the Private setting in the Tools menu. The only way to resume normal browsing is to change the about:config back to its original False setting. Doing this restores your last previous non-Private browsing session, tabs and all.
One of the big disappointments is that the new tab-switching behavior has been removed. As a serial tab abuser, I thought it was extremely useful that FF3.1 would have a visual representation of the tabs I was jumping between. Hopefully, this feature will be at least rolled out as an add-on, but I've been known to abuse those, too.
Firefox 3.1 beta 2 will not be the final tester update. According to Mozilla, it will be followed by at least one more beta release.