The Dragonspeak is the dragon language used in Final Fantasy XIV.
Behind the scenes
The dragon language was created by Michael-Christopher Koji Fox, the Translation Director and English Localization Division, before the original Final Fantasy XIV launch, when Ishgard was still slated to be a part of 1.0. At that time, all the beast tribes were going to have unique tongues (which would be voiced), the Echo simply "translating" those languages for the player in the dialogue windows. Fox presented the idea to Kenichi Iwao, the Lore Master at that time but unfortunately, by the time Fox had completed his work, it was apparent that Ishgard wasn't going to make it in 1.0.
In patch 2.X, the dragons were going to make an appearance to setup the way for the game's Ishgard-themed expansion, Fox re-presented the Dragonspeak to the new Lore Master and was granted permission.