Especially for new runners. You're breathing backwards, and you need to learn to stop heaving with your chest, and instead breathe by expanding your diaphragm (the muscle right below your lungs and ribs). Breathing backwards heavily for any extended period at an increased rate will make you feel like your chest is going to pop. People with bad posture tend to have this problem most often, so you also need to work on standing, sitting, and walking with your back straight and your head up if this is an issue.
Watch how you breathe. When a lot of people hyperventilate (the same breathing pattern is simulated when running for a lot of new runners, even if they don't do it when relaxing), they can't get the breath in because when they breathe in, they depress their stomach, and raise the chest. When breathing out, they relax their stomach and lower the chest. This causes your lungs to be constricted when you're trying to draw in air. Practice breathing in and feeling it all the way into your stomach, because it should pooch out from filling with air. If you're doing it right, you can test by placing your hands around your natural waist and take in a big breath. Your fingers and thumb should move further apart and your back should also expand with your stomach as you breathe in.
To make this a habit when running, breathe in through your nose, and out of your mouth on rhythm with your pace. I usually breathe in for 3 steps, and out for 3. You might get side cramps at first, (and if you are working on posture) and your back might be sore, but they will go away as your lungs get stronger and are able to take in more air and your posture improves. You will also notice increased endurance and overall better calorie burn because you're providing your body with enough oxygen to be efficient.