PS3 owners of Final Fantasy XIV will be able to upgrade to the PS4 version for free starting today. The upgrade will be available until "at least" the end of the year, giving players time to upgrade their PlayStation consoles. However, do note that once the upgrade is performed, you'll lose access to the PS3 version.
In order to upgrade, you'll first need to log into MogStation with a Square Enix account. Then, you'll see a banner to upgrade from PS3 to PS4 on the account services page. You'll have to agree to the terms and click "Upgrade" to get a PlayStation Network code.