Your muscles main source of fuel is glycogen or fat. In order for muscles to consume this fuel, oxygen has to be present to break it down and convert into ATP (adenosine triphosphate), a chemical compound which is the form of energy that muscle cells need in order to work.

The production of ATP leaves byproducts in the form of carbon dioxide and lactate. The carbon dioxide laden blood is pumped back into the lungs, where CO2 is extracted and exhaled into the atmosphere. The lactate is also removed from the muscles by the venous blood flow and delivered to the liver where it is converted into glucose.

As the intensity of your running increases your muscles will be working harder and will need to metabolize glycogen or fat at a proportionately faster rate. Your system will have to increase the input of oxygen to achieve this. Hence the heavy breathing or gasping for air during and after hard workout.

When you are running, your heart should be beating at between 60% and 80% of its maximum rate, fluctuating according to the intensity of your running. If you keep it beating at this level, you'll be pushing your heart hard enough to reap the benefits of having a good workout but not so hard that you'll be putting it under stress.

Your heart rate is not necessarily related to what speed you are running at because your heart will be registering all sorts of extraneous factors -- how stressful a day you've had, what the temperature is, how much coffee you've put a way, if you're sickening for something -- and will adjust itself accordingly.