Does Firefox support private Web browsing?

First off, yeah, I'm a parent, so let me say that if you're going places that your parents prefer you didn't, then maybe you should respect their wishes or work with them to create the trust that will let them allow you to visit the site or sites without any duplicitousness?

No? You'd rather just hide your tracks? Well, I can reluctantly show you one very cool Firefox feature that'll let you do exactly what you seek, use it to surf the Web without leaving any crumbs or clues...

Launch Firefox and go to Preferences --> Privacy. You'll see this:

Look closely near the bottom and you'll see "Always clear my private data when I close Firefox". That's what you want to set, so check that, then click on "Settings..." immediately adjacent. Now you'll see that you can specify which data should be wiped:

Check all of these and then each time you quit Firefox you'll leave no trace of cookies, password settings, browser history, download history, saved forms, etc etc etc. No trail.

That's what you want, right? Good luck.