If you're moving FATS from one machine to another, the first thing is to figure out where your data is stored.
You can do this by going to Utilities | Advanced | System Setup and looking for the "database location" field.
If the database is on a file server, you don't need to worry about backing it up, it will still be there. (If you have multiple people sharing the data file, this is likely the case, unless you are sharing out a folder on your machine.)
If the database is on your local machine, it will probably be under c:\Program Files (x86)\FATS6.
There is one caveat, however. If you are on Vista or later, you will see a button called "compatability files" when you navigate to that folder. Click this button. This takes you to a secret directory in your application folder. The secret folder is hidden under c:\Users\<your username>\AppData\Local\Virtual Store\Program Files\FATS6.
The "real" files should be in this folder. You might want to check both folders for the following files:
FATS6Data.mdb (holds all your data) (take the one with the most recent time stamp)
ComputerDirectionsCompany.mdb (this is the license key. It's probably also on your FATS installation CD, but you can use this one as well.
Copy FATS6Data.mdb to anywhere you would like on the new machine (or put it on a shared network folder if you would like).
Drop ComputerDirectionsCompany.mdb on your desktop. Once you've used it, you can delete it from the desktop.
When you launch FATS, it will ask if you want to connect to a database, choose the FATS6Data.mdb that you put in the location you chose.
It will also ask you for a Company Name File. You can navigate to the one on your desktop. Once it copies this file where it wants it, you can delete the one on the desktop. Some customers store this file on the network in the same folder as their FATS6Data.mdb for safe keeping.