Everyone is always saying that it's a ponytail. Well it isn't. What Zuko used to have was called a "Topknot". They are very similar in most respects, but with some key differences:
1.) A ponytail starts at the base of the neck. A topknot starts at the scalp.
2.) With a ponytail, the rest of the hair on th head remains. With a topknot, all the hair (except for the topknot itself) is shaved off.
3.) In "ATLA", a ponytail is just a hair style. A topknot is a denotion of the royal family of the Fire Nations (in males, at least).