How does the circulatory system work with the respiratory system?

The respiratory and circulatory systems work together to deliver oxygen to the tissues and to remove the carbon dioxide byproduct of metabolism from the body. Respiration that occurs within cells and breaks down energy sources such as glucose to make ATP use oxygen and produce carbon dioxide. Oxygen must continuously be delivered to these cells in our tissues in order to keep respiration going and carbon dioxide must be removed to avoid issues like acidosis.  

At the systemic tissues carbon dioxide is at a relatively high partial pressure because it is being produced and oxygen is at a relatively low partial pressure because it is being used by the cells. When freshly oxygenated blood arrives from the left side of the heart to the capillaries surrounding the tissues, it has a relatively high partial pressure of oxygen and low partial pressure of carbon dioxide. Because of this and because gases diffuse from high partial pressure to low partial pressure, oxygen will diffuse from the capillaries to the tissues and carbon dioxide will diffuse from the tissues into the capillaries.  

This blood is then taken back toward the heart through the veins of the body and eventually enters the right atrium through the superior and inferior vena cava. At the same time blood is entering the left atrium from the pulmonary veins coming from the lungs. Both atria then contract, sending the blood they contain through the atrioventricular valves and into the ventricle on the same side of the heart. When the ventricles contract the pressure closes the atrioventricular valves and the blood enters the pulmonary trunk from the right ventricle and the aorta from the left. When the blood entering the aorta will then deliver more oxygen to the tissues, the blood from the right side will enter the pulmonary circulation.

From the pulmonary trunk, the blood will go into either the right or left pulmonary artery and then will enter the capillary beds around the alveoli (air sacs) of the lungs. Diffusion will happen here as well. Within the alveoli there is air that has recently been inhaled. It has a relatively high partial pressure of oxygen and relatively low partial pressure of carbon dioxide due to their concentration in the atmosphere. The blood that has traveled to the area from the right side of the heart was depleted of oxygen at the tissues of the body and has a relatively high partial pressure of carbon dioxide as it was picked up at the tissues by the blood. Because of these differences in partial pressure, oxygen gas will diffuse from the alveoli to the blood in the capillaries and carbon dioxide will diffuse from the blood into the alveoli so that it can be exhaled. Then, finally, this blood goes back to the left side of the heart through the pulmonary veins before it is again sent out to the tissues once again. 

Note that diffusion only occurs at the capillaries when we are talking about the circulatory system. This is because they are lined with thin, flat cells (simple squamous epithelium) so diffusion occurs readily. The walls of other vessels are too think for gasses to diffuse effectively.