How does the body respond following a meal to ensure a relatively constant blood sugar level?

After a meal blood sugar levels will rise. In the Islets of Langerhans beta cells detect hyperglycaemia and secrete insulin, while alpha cells stop releasing glucagon.
The insulin moves in the blood stream to the liver where it binds to receptors on a hepatocyte. The binding of the insulin to a receptor causes a Glut-4 vesicle to move to and fuse with the cell surface membrane by exocytosis. This increases the permeability of glucose into the cell. Glucose is then removed inside the cell by either glycolysis to produce ATP or conversion to Glycogen in glycogenisis.