Nightmare Moon was an alicorn with a jet black coat, her mane was an ethereal night sky, filled with stars that flowed on its own, much like Celestia's own mane. She had teal, dragonesque eyes and her cutie mark was a cresent moon on a purple, twilight sky, almost identical to Luna's own cutie mark, bar the color change.
Nightmare Moon's first appearence is NOT her introduction, she makes a brief camo appearence in Twilight's hourglass around four minutes and forty seconds in.
It is unclear if "Nightmare Moon" is an Entity which possessed Luna or not; the show seems to lean that Nightmare Moon is Luna's alternate personality but the comics state that Luna was corrupted during her time on the moon.
Long ago, Luna and Celestia were both the rulers of Equestria, a land inhabited by ponies. Every morning, Celestia would raise the sun, and every evening, Luna would raise the moon. Together, they kept the balance perfectly. One day, however, Luna grew jealous: the ponies would be awake to enjoy the day and love, admire, and appreciate Celestia's work, but would always sleep through her far more beautiful night. One day, Luna turned on Celestia, saying she should be the only princess in Equestria, and raised the moon before the bitterness in her heart manifested and transformed her into Nightmare Moon.