ponytail vs hairdo

ponytail and hairdo both are nouns.

Word	Noun	Adjective	Verb	Adverb
ponytail	Yes	No	No	No
hairdo	Yes	No	No	No

As nouns, hairdo is a hypernym of ponytail; that is, hairdo is a word with a broader meaning than ponytail:

ponytail: a hair style that draws the hair back so that it hangs down in back of the head like a pony's tail
hairdo: the arrangement of the hair (especially a woman's hair)

Other hypernyms of ponytail include coif, coiffure, hair style, hairstyle.

ponytail (noun)	hairdo (noun)
a hair style that draws the hair back so that it hangs down in back of the head like a pony's tail	the arrangement of the hair (especially a woman's hair)