Breathing in Yoga
Breathing while practicing yoga poses, is one of the essential parts of doing yoga.

For the record, you should breathe when you're not practicing yoga too!
Once you become familiar with some of the key yoga poses you can start observing how you are breathing while in the poses. You can also use breathing exercises (pranayama) when sitting still to deepen your breath and calm the mind.
Vinyasa means linking breath with movement
What makes physical yoga different to other forms of exercise is the focus on your breath with every movement. The philosophy behind this is that as long as you are consciously breathing while moving, you are doing a vinyasa yoga practice. This is helpful to keep in mind when beginning yoga or practicing some difficult poses. This is because the in yoga we prioritize having a steady breath over getting your body into a particular yoga pose.
For example, imagine attempting a challenging backbend like the wheel pose (urdhva dhanurasana). It might feel tempting to use some force in order to achieve the full expression of the pose. You might find in the process of doing the pose that you have started to lose your steady breathing pattern. Instead, you want to try to primarily focus on the breath, and to go only as far into the pose that you can while still maintaining your regular breath.
In essence you want to find your limit, and stay a little below it, so that you can explore the pose while keeping your breath flowing smoothly. Naturally all of us have a tendency to push ourselves either a bit too much, or a bit too little. One of the goals of yoga is constantly searching for the balance somewhere in the middle.
Slow breathing calms the mind
The role of a steady rhythmic breath while moving through different yoga poses can help to create an internal rhythm to guide you through your yoga practice. By keeping a regular rhythm to your breath and movement, it can help over time to focus your thoughts and calm your mind.
In addition to watching your breath, while moving through different yoga practices, you can also practice formal breathing exercises called pranayama.
Sama Vayu Pranayama
One way to do this is to sit in a comfortable crossed leg position, and place one hand on your belly, and the other on your chest. First take a few natural breaths and become aware of the speed and depth of your breath. Then, you can begin to slowly allow your breath to deepen and lengthen. Let the belly expand and chest lift on the inhale, and belly contract and chest drop on the exhale. See if you can do this for 20 – 30 breaths. This breathing exercise (pranayama) is called sama vayu, and is great to practice in order to help unlock a deep natural breathing pattern.