A barber is a person who cuts men's hair and shaves/trims beards as their occupation. It is not a gender-neutral term - barber is typically used to refer to a male who cuts the hair of other males. A hairdresser is a person who cuts and styles hair as their occupation and is not gender-specific. A barber is also a hairdresser but a hairdresser is not always a barber. Generally most the clientele of a hairdresser are women but men also have their hair cut here. In my experience, most men will go to a barber to have their hair cut. It is clear when the place is a barbers and is normally clearly labelled on the shop front. A barber normally does a straightforward, quick, and cheap cut/trim. I go to a hairdressers now for convenience and it has been more expensive but they have paid more attention to styling my hair and wash it too. Establishments which only cut women's hair are sometimes called hairstylists, cosmetologists, beauticians or makeover artists.

They are both common, but different businesses. Barbers generally only cut men's hair. They are usually cheaper than other hairdressers. They offer other male-oriented services like shaving. Hairdresser is a more general term that could probably include barbers. But when I think of a hairdresser I think of someone who cuts both men and women's hair, and therefore can do more complicated cuts/styles, and for a higher price. They might not provide face-shaving services.