The use of the word "rank" for Navy enlisted personnel is incorrect as the word "rank" is only used for officers. The enlisted term is "rate." The rating badge is a combination of rate (pay grade as indicated by the chevrons) and rating (occupational specialty as indicated by the symbol just above the chevrons).
The insignia on the left represents a Petty Officer First Class (the rate) who is a Boatswain's Mate (the rating). A rating badge is worn on the left upper sleeve of all uniforms in grades E-4 through E-6. Chief Petty Officers (E-7 through E-9) wear collar devices on their white and khaki uniforms, and rating badges on their Service Dress Blues.
Navy officers wear their rank devices in different places on their uniforms, depending upon the uniform. The three basic uniforms and the type of rank devices are: khakis (a working uniform) - pins on the collar; whites - stripes on shoulder boards; and, blues - stripes sewn on the lower sleeve. Shoulder boards are also worn on bridge coats and reefers.
Officers in the Navy are either Line officers or Staff Corps. Among these are also Limited Duty Officers and Commissioned Warrant Officers. Staff Corps and Commission Warrant Officers wear Insignia in place of the Line officer's star.
US Navy Officer Ranks
US Navy Enlisted Rates