The use of the word "rank" for Navy enlisted personnel is incorrect. The 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 here 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.
http://www.navydads.com/forum/topics/the-difference-between-rate-and-rating
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