The launcher stores launcher versions, instances, and your configuration in only one folder:
To restore the files on a different computer, just copy the backup to the same place.
If you are moving from Windows to Mac OS X, vice versa, or to/from Linux, you will need to move your files to the new location. You can just run the launcher on the new computer, see where the files are put, and then replace them with your backup. Remember to right click every instance and click "Force update" because switching computer types means different files are needed.