The hairdresser's is the shop you go to when your hair needs cutting. Go to the hairdresser's, book the hairdresser's, at the hairdresser's.
The apostrophe is doing real work, since it means the hairdresser's shop with the shop left out, like the baker's or the doctor's. It takes the rather than a in almost every sentence.
1nounhairdresser'sThe place you go when your hair needs cutting.
“She was at the hairdresser's when the call came.”
hairdresser's is a shop.
Also accepted in games: salon, barbershop, hair salon