Sometimes, there's a reason one will breathe through their mouth and that is perfectly normal. When you are stopped up with a cold, sometimes the only way to breathe is through your mouth. When you are out of breath, you may find yourself breathing throug your mouth as well. When breathing through your nose, the lungs get a little break time because you take in less air and let it out slower.
This is a more natural method for the majority of the time. When breathing through your mouth, the lung fill up more and faster which causes them to work harder. All said and done, it is best to breathe through your nose when you can and through your mouth when you need to but it is definately possible to do both although not at the same time.