The breath is the thread that ties your practice together. Breathing in and out through the nose is your body's natural way to help you stay present, calm, focused and in control. Focusing on your breath helps disconnect you from the external world and from your worries and obligations, enabling you to get the most benefit from your practice.
Being mindful of your breath also helps you pay attention to your body as you move into, out of, and through poses, so that you don't hurt yourself. Apart from the first and last breathing exercises, breathe in and out through the nose. Breathe calmly, smoothly and continually – never holding your breath. Inhale as you move upward/expand/open/lengthen; exhale as you bend forward/fold/twist/relax.
Use the 80-20 breath while holding poses, to help yourself move deeper into the pose (click "here" for detailed info on 80-20 breathing). Pay attention to your teacher's instructions. Think about the breath at all times, and in time your breath control will improve enabling a stronger and deeper practice.
If your breathing isn't relaxed, your body can't relax into the postures. If your body isn't relaxed, your mind can't relax. And if your mind isn't relaxed, you can't draw the full benefits from your yoga practice.