In my experience that usually happens when it crashes out halfway through downloading a patch and deleting the temporarily download files seems to fix it.

To do this, close the launcher if it's running then go into your Documents directory (wherever that is) and navigate through to the downloads folder like so:

Documents -> My Games -> FINAL FANTASY XIV - A Realm Reborn -> downloads

In this directory you should find a ffxivupdater program (don't touch/delete this!) and an ffxivpatch directory. Delete the ffxivpatch directory (and anything inside it). That's the patch(es) it's downloading so deleting the directory forces the launcher to redownload that patch the next time you start it.