Each Firefox installation has one or more profile directories; the directory names are random characters (different on each installation) with a meaningful file extension, so after you copy a profile directory to a different computer, you need to tell it about the profile directory.
I'm no expert, but I migrated Thunderbird (similar issues) and Firefox from one laptop to another a couple of weeks ago. A few minutes on Google will turn up instructions on migrating profiles to a new installation and convincing the new install to take on the old personality.