You need the two files signons.sqlite and key3.db to transfer or backup the passwords.
The key3.db file stores the encryption key and the correct version of this file is required to be able to decrypt the passwords.

See also:

• https://support.mozilla.org/kb/Recovering+important+data+from+an+old+profile
• http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox

You can also look at this extension:

• Password Exporter: https://addons.mozilla.org/firefox/addon/password-exporter/

What are the fully-qualified locations of the two files you mention?

You can use this button to go to the Firefox profile folder:

• Help > Troubleshooting Information > Profile Directory: Show Folder (Linux: Open Directory; Mac: Show in Finder)

• http://kb.mozillazine.org/Profile_folder_-_Firefox

Main profile folder:

• C:\Users\<user>\AppData\Roaming\Mozilla\Firefox\Profiles\<profile>\

Cache folder location as shown on the about:plugins page:

• C:\Users\<user>\AppData\LocalMozilla\Firefox\Profiles\<profile>\

The "Application Data" folder in XP/Win2K and the "AppData" folder in Windows Vista and Windows 7 and later versions are hidden folders.

• http://kb.mozillazine.org/Show_hidden_files_and_folders

Thanks!