The importance of proper breathing has been stressed and mentioned many times. But how do you start to retrain yourself in proper breathing if you have never ever paid attention to it before? 
Most yoga asanas help to deepen the breath and encourage breath awareness. So it might be a good idea to go through a few yoga positions before sitting/lying/standing still to focus entirely on the breath. 
It also would be useful to mention that unless otherwise specified we breath through our nose. 
Lets first focus on the individual phases of full yoga breath starting with abdominal breath. Lie down on your back, bend the knees. Put one of your hands on your abdomen and the other one on your chest. Notice the movement of your breath underneath your hands. Consciously draw your breath into your abdomen so that you can feel the expanding movement underneath the hand on your abdomen. Try to keep the other hand on your chest still. Comfortably expand your abdomen with each inhalation and feel it collapsing on each exhalation. Feel your breath not only at the front of your body but draw your inhalations to the sides and the back as well. Keep up for couple of minutes or more. Abdominal breath calms the body and the mind, improves our energy levels and positively stimulates all the organs in the abdominal cavity. 
Stay in the same positions for the next phase - the rib cage breath. Draw your inhalations a little higher now, into your rib cage. It is the hand on your chest which will feel more movement. Try to keep the hand on your abdomen still. In order to feel the expansion of your rib cage to both sides and the back you may place both hands on either side of your ribcage and feel them moving further apart on inhalations and back closer together on exhalations. This breath is a little more energising but still calming with positive effects on blood circulation and the organs in thoracic cavity. 
Clavicular breath is the most shallow one with only about 10% of oxygen inhaled with this part. Normally we use the very top of our lungs when we are in the middle of strong emotions. Our breath fastens and switches to the shallow clavicular breath as other parts of the body are tense and tight. Although not desirable on its own, it is still an important part of the full yoga breath. In the same position on your back, you may try to put your hand on your collar bones and simply try to draw your inhalations there. You want to feel gentle rising movement of your collar bones but shoulders don't need to rise up at all or only very little. You won't be able to keep your rib cage still, that is fine, just focus on the very top of your chest. 
And now all you need to do is to join these three phases of the full yoga breath together! 
You may like to keep your hands placed on your abdomen and chest or you can use the movement of the arms to help you make the full yoga breath smooth and flowing.
On one inhalation fill your abdomen first as you start lifting your arms, then fill your ribcage with your arms pointing straight up and then draw the last bit of the air into the very top of your chest while your arms rest behind your head. Exhale in the same order and reverse the movement of your arms. 
Have fun and remember that only three minutes a day of full, mindful breathing makes huge difference in your wellbeing.  

 I'm pretty sure you are breathing at the moment. But how? Well, in and out you may say. Yes, that is a good start. But do you notice yourself breathing? Do you know how fast you breathe? Which part of your body moves when you breathe? Do you breathe into your abdomen or your chest? Breathing matters! Although it is an automatic process for most of us some breathing awareness goes long way in improving overall wellbeing, especially our energy levels. 
Pranayama is a big part of yoga and means breath awareness and control. Prana means breath, energy and life. Prana is life energy and is present in everything that is alive. It unites the body, mind and spirit and any obstructions in its flow affects all processes within our bodies. Breathing techniques are one of the best ways to bring more energy into our bodies and minds and feel more 'alive'.
There are many breathing techniques available and it is not important or necessary to learn them all. It is much more effective to practise two or three pranayama techniques only but every day. 
Start with breath observation. Sit  on the floor, in a chair or lie down on your back in savasana. Watch your breath without controlling it, only as a passive observer. 
First and most important breath to master is Full Yoga Breath - Dirgha Pranayama. This breath uses the lungs to their full capacity and has three parts - abdominal, thoracic and clavicular breath. Proper breathing must start with the diaphragm - a big flat muscle which divides abdominal and thoracic cavities. When we breathe in the diaphragm moves down and pushes our abdomen out. If our abdomen doesn't blow out a little when we breathe in it suggests that we don't use our diaphragm properly. Incorrect posture and mental tension are main reasons for not using our diaphragm as we should. 
This abdominal breath is only one part of full yoga breath. Next we need to open and widen our rib cage and, in step three, draw our breath and open the very top of our chest as well.  We complete all three parts on one inhalation and exhale in the same order. Don't worry if these three stages feel a bit disjointed at first. With regular practise full yoga breath will feel like a gentle wave moving through your upper body.