Posture isn't important just at grandma's house. Proper running posture can increase your lung capacity and stride length, improving the efficiency of oxygen exchange and making you go faster. Your posture also sets the tone for the rest of your body. The position of your torso impacts your head, shoulders, arms and hips -- all critical components to your running form. Ideal running posture means looking straight ahead, not at the ground. This naturally straightens your back so that it's vertical and perpendicular to the ground. It also opens the lungs and lets you draw deeper breaths, oxygenating blood more with every breath. Your shoulders should be square, but not robotic. Keep them level and relaxed, minimizing the amount of side-to-side swing that can cause the rest of your body to compensate for being tilted off balance with every step.
One trick for improving your posture is to try to take a deep breath in through the nose and out through the mouth while you're running. This will cause you to extend and become more upright. If you can breathe more easily and comfortably maintain this posture, you were probably slouching to start with.