Her name is Faron, the Water Dragon.
Faron the Water Dragon is a large, blue reptile-like deity who resides in the Faron area in Skyward Sword. She is one of the "Three Dragons" that guard the land below The Sky. The other dragons being Lanayru the Thunder Dragon, and Eldin the Fire Dragon. Throughout the game she stays very skeptical of Link's abilities and is a very important character. She is also very strict about manners and will often correct the hero for being impolite. After completing the Silent Realm in Faron Woods, Link will be rewarded with the Water Dragon's Scale. An item that belongs to the Faron herself, it allows him to swim underwater and do spin-attacks. Faron can be found deep within Lake Floria. Furthermore, she reveals the path to the Ancient Cistern, the fourth dungeon of the game. She also teaches Link the first part of the Song of the Hero later on in the story. To do this, Link must obtain the Tadtones as instructed by Faron.