The respiratory pigment in red blood cells that carries oxygen is called hemoglobin. Hemoglobin binds to oxygen in the lungs and transports it to tissues throughout the body. This protein's structure allows it to pick up oxygen in high-concentration areas and release it in areas where it is needed. Hemoglobin also assists in transporting carbon dioxide, a waste product, back to the lungs for exhalation, thus playing a vital role in maintaining respiratory and metabolic homeostasis.