These add-ons can be a great help by backing up and restoring Firefox

FEBE (Firefox Environment Backup Extension) {web link} FEBE allows you to quickly and easily backup your Firefox extensions, history, passwords, and more. In fact, it goes beyond just backing up -- It will actually rebuild your saved files individually into installable .xpi files. It will also make backup of files that you choose.

OPIE {web link} Import/Export extension preferences

Is this something that I would install on my current computer or something I would install on the new computer if mine crashed - I'm not understanding how the new computer would know it was me if I downloaded Firefox onto it.

Those programs will save your Firefox information in folders you choose. If needed, you can copy those folders to a thumb drive.

I had heard of this, but it took me a while to find it.

http://portableapps.com/apps/internet/firefox_portable Mozilla Firefox, Portable Edition

Mozilla Firefox®, Portable Edition is the popular Mozilla Firefox web browser bundled with a PortableApps.com Launcher as a portable app, so you can take your bookmarks, extensions and saved passwords with you.

Thank you very much

Glad to help out.