Pranayama are breathing exercises traditionally used in Yogic practices that work on vital energy (prana). They can be potently used to create calmness or energy depending on the breathing exercise. All breathing exercises should be done with some supervision as they can cause unwanted side effects if done incorrectly or with certain medical conditions.
Below are instructions for two breathing techniques that you can also find elsewhere. There are variable versions of these. A good book to reference is The Healing Power of the Breath: Simple Techniques to Reduce Stress and Anxiety, Enhance Concentration, and Balance Your Emotions by Richard Brown, MD and Patricia Gerbard, MD.
The following example is from Practical Yoga Psychology by Dr Rishi Vivekananda:
This practice is used in yoga therapy to soothe the nervous system and calm the mind. It has a profoundly relaxing effect at the psychic level, and as it helps to relieve insomnia, it may also be practiced lying down just before sleep.
We sit in any comfortable position, with the eyes closed, and the whole body relaxed, become aware of the breath in the nostrils, and allow the breathing to become calm and rhythmic.
After some time we become aware of the throat, and try to feel or to imagine that the breath is being drawn in and out through the throat and not through the nostrils; almost as if inhalation and exhalation are taking place just through the throat.
As the breathing becomes slower and deeper, we gently contract he glottis so that a soft snoring sound, like the breathing of a sleeping baby, is produced in the throat. Both inhalation and exhalation are long, deep and controlled. It is essentially yogic breathing while concentrating on the sound produced by the breath in the throat.
The sound of the breath should not be very loud. It should just be audible to the practitioner, but not to another person unless they are sitting very close.
Alternate Nostril Breathing (Nadi Shodhana)
This technique induces tranquility, clarity of thought and concentration, and is recommended for those engaged in mental work. But most importantly, it also balances the flow of the breath through the two nostrils, so it balances the activities of the two sides of the brain, causes sushumna nadi to flow and so readies one for meditation.
We hold the fingers of the right hand in front of the face, resting the index and middle fingers gently on the point between the eyebrows, with both fingers relaxed. The thumb is then covering the right nostril and the ring finger is covering the left. During the practice, these two fingers are the ones that control the flow of breath in the nostrils by alternately pressing on one nostril, block the flow of breath, and then the other. The little finger is comfortably folded.
We sit in a comfortable meditative posture keep the head and spine upright, body relaxed and the eyes closed. We practices yogic breathing for some time, then place the hand over the face as explained above, place the left hand on the knee, and close the right nostril with the thumb. Then we inhale and exhale through the left nostril 5 times at a normal rate, being aware of each breath, making sure that there is no sound as the air passes through the nostrils After 5 breathes we release the pressure of the thumb on the right nostril and press the left nostril with the ring finger, blocking the flow of air, and inhale and exhale through the right nostril 5 times. The we lower the hand and breathe 5 times through both nostrils, and this completes one round. It is usual to practice 5 rounds or for 3-5 minute.
(Of note, there are other variations of Alternate nostril breathing)