Both the pulmonary circuit and systemic circuit are parts of the cardiovascular system, responsible for transporting blood through the body.
The pulmonary circuit pumps blood between the heart and the lungs. This is necessary to keep the blood oxygenated and also to keep blood pressure up; without the pressure from the heart, the blood flowing out of the lungs would not have enough pressure to continue through the body. Because of this, the blood travels through the lungs and then returns to the heart, where it is pumped out through arteries and then through the rest of the body.
The systemic circuit is the means by which the blood travels throughout the body, discharges its oxygen load, and then returns to the heart. The systemic circuit contains veins and arteries; the arteries pump oxygenated blood away from the heart and the veins transport the newly de-oxygenated blood back to the heart.
The pulmonary circuit is on the right side of the heart, and forms a closed loop between the heart and lungs. The systemic circuit is on the left side of the heart, and forms a closed loop from the heart through the arteries and veins.