The respiratory system and the circulatory system work nonstop. The respiratory system is dependent on the circulatory system and vice versa, as the heart cannot work without the oxygen it receives from the lungs and the respiratory system cannot function properly without the help of the circulatory system, because the circulatory system transports the oxygen and blood to the entire body and thus ensures that the body is working properly.
The main organ of the respiratory system, the lungs, and the main organ of the circulatory system, the heart, connect these two systems.
When we inhale, the air, which contains oxygen and other nutrients, travels through the lungs (specifically through the bronchi) and reaches the alveoli, which are surrounded by small capillaries or blood vessels. The oxygen is then transferred from the alveoli into the red blood cells (erythrocytes); the hemoglobin, an iron-containing protein in the red blood cells, carries the oxygen through the body and to all of the organs and tissues.
Carbon dioxide, which is a waste product of the process of cellular respiration, is released from the blood; it travels from the blood cells into the capillaries and then to the lungs, from where it's released when we breathe out.
The blood that is rich with oxygen (oxygenated blood) travels from the lungs, though the pulmonary vein, and into the left ventricle of the heart, which pumps the blood into the arteries. The arteries (which are made of arterioles and the capillaries I've mentioned) then carry the blood around the body and supply it with oxygen. The blood that contains carbon dioxide travels through the pulmonary artery and the veins to the right ventricle of the heart, which pumps the deoxygenated blood into the lungs; from there, the carbon dioxide and other waste products leave the body as we exhale.