Standard Private Browsing with Firefox
Normally, if you want to keep your online movements under wraps from other users of your computer, you would use the Private Browsing mode in Firefox. This tool does not make you anonymous on the web, but does prevent the pages that you are visiting from being recorded in History. Clearly this is quite useful, and is easily activated via Tools > Start Private Browsing or the CTRL+SHIFT+P keyboard combination.
Disabling the Private Browsing mode is simply a case of repeating the above action, but what if you want to leave it on permanently, by default?
In this case you would use the Options… screen on the Tools menu. Via the Privacy tab, select Use custom settings for history and then check the Automatically start Firefox in a private browsing session option. Click OK to exit, and the browser will then automatically launch without saving any history.
Note, however, that this might make finding some interesting material a little tougher later on. If you have concerns that this may occur, you can use the standard browsing mode combined with the following steps.