A barber is a person who shaves beards and cuts hair.

Literal

Barber [topic-は] person [possessive-の] beard [object-を] shave, haircut [object-を] do person is.

理髪師 (barber) is the formal term for the profession. ひげをそり (shaves beards) and 散髪をする (cuts hair) define the job. A definitional sentence.