In your Documents folder you'll find: My Games\FINAL FANTASY XIV - A Realm Reborn\

Make a copy of those folders (you'll need to move them back exactly as they are after reinstall). They're not "critical", but if you don't you'll have to accept the EULA again and create all your hotbats/gearsets again, so it's best to make sure you have that.

Then in wherever you installed FFXIV you'll likely find a SquareEnix\FINAL FANTASY XIV - A Realm Reborn folder. Just copy all of that somewhere else, exactly like it is.

You don't actually need to reinstall the game if you don't want because it doesn't create any registry entries and will happily run just by copying it back, but if like me you want completeness and it to create the shortcuts etc then that's simple enough:

Download/install the launcher (109 Mb) from: http://www.finalfantasyxiv.com/playersdownload/na/

(the eu page links to exactly the same installer, it's the actually account details you sign in with that are region locked, not the game itself).

But after installing the launcher don't do anything else (if you run it, it'll start creating your document files and it'll want to download the game etc).

At this point after having installed the launcher move all your files back again. You probably won't have a Documents\My Games folder on a clean install, but if you create/move the files back exactly like they were inside the documents folder before, they'll be fine). It doesn't matter if your documents folder has changed location, only that the subdirectories/files inside it are in the correct places.

Then move the SquareEnix\FINAL FANTASY XIV - A Realm Reborn big game files back over as well. Overwrite any existing files (because they'll be old versions of the launcher etc).

And that's it! This is exactly how I installed it on my netbook without having to redownload it all and you'll have a fully reinstalled game with only minimal downloading