The respiratory system is the set of organs responsible for getting oxygen into your body and getting carbon dioxide out of your body. Primarily, this concerns the lungs and the alveoli, where the gaseous exchange occurs.
The cardiovascular system is the set of organs responsible for delivering blood to every cell in your body. This would include the heart, the arteries, the veins, and the capillaries.
The heart moves the blood every time the ventricles contract. Blood is sent to the lungs for two reasons: 1) to get rid of waste products, mainly carbon dioxide, and 2) to pick up life-giving oxygen. The blood dumps it's load of carbon dioxide in the capillaries of the alveoli in the lungs. You exhale the carbon dioxide, while in the next instant you inhale a fresh load of oxygen. The oxygen diffuses across the alveoli's capillaries into the blood, which is returned to the heart. The heart then sends the freshly oxygenated blood throughout the network of arteries, capillaries, and veins.