Early Access for Final Fantasy XIV: Heavensward is now live, and players can currently register for it using the link listed in the official FFXIV tweet listed below.
— FINAL FANTASY XIV (@FF_XIV_EN) June 19, 2015
The FFXIV website that the link will bring you to goes over exactly how to sign up for Early Access, as well as how to use the special Early Access codes.
A launch trailer for the expansion was also released recently, and that can be seen at the top of this post. Accompanying the trailer were the full patch notes for Patch 3.0, which can be seen here. A huge amount of new areas, quests, and flying mounts have been added in the patch. Early Access players can take advantage of the new updates today, but everyone else will have to wait until June 23 to play the Heavensward expansion.