Controlling or extending the breath (prāṇāyāma) is, or should be, a big part of any yoga practice. As far as we know, it has been central to the practice of yoga since its earliest times and it is a foundational practice for haṭha yoga (pronounced "hut-ta", sometimes "hat-ta"). The earlier Yogasūtra-s offers it as an important practice (e.g. YSII.49-51).
Most of us in the West probably know yoga in one form or other of haṭha yoga, even if we don't know it as that. I suggest this is the case because we all get to learn and practise yoga postural work called āsana or yogāsana and āsana forms part of haṭha yoga, albeit of lesser importance in the modern context than prāṇāyāma. I find it interesting that in Western yoga, āsana-s are the defining practice of modern yoga. However, what distinguishes pre-modern yoga was prāṇāyāma.
I think the time has come for us to re-discover the power of prāṇāyāma. Why? The short answer is that we are more than just a biological computer (brain) in a biological machine (body). Āsana-s help our bodies to remain strong, flexible and functionally efficient. Prāṇāyāma, on the other hand, has a hugely positive impact on the way our Autonomic Nervous System ('ANS') works.
The ANS is divided up into several components. For instance, it helps us deal with perceived danger, for example, the well known fight or flight response, and it can promote rest and digestion. But not both at the same time. Modern living causes chronic dis-regulation within the ANS and the ancient and beautiful techniques of prāṇāyāma restore the proper functioning of our ANS.
Prāṇāyāma is a compound word made up of pṛāṇa (meaning something along the lines of 'life-breath') and āyāma (meaning 'extension' or 'control'). Prāṇāyāma can mean, therefore, the lengthening of or the control of the our life-energy.
When our ANS is in a state of dis-regulation it feels like our life-energy, prāṇa, is weaker than it should be and not where it should be, i.e. in our bodies. There is a wonderful haṭhayoga text, the Yoga Yājñavalkya Samhitā, which explains that the prāṇa for one whose mind is troubled, restless or confused lies outside her body.
One purpose of prāṇāyāma, maybe the purpose, is to bring prāṇa (and therefore our life-energy) deep into the body. So deep, in fact, that it enters the Suṣūmṇā nāḍī, which runs along the spine.
There are a number of different methods for practising prāṇāyāma that are mentioned in classical yoga texts like Yoga Yājñavalkya Samhitā ('YY') and the Haṭha Yoga Pradīpikā ('HYP'). The Yogasūtra-s ('YS') gives some very good principles for prāṇāyāma.
Qualities of prāṇāyāma
The Yoga Sūtra-s probably gives us our best guide as to what we should be aiming for when we practise prāṇāyāma. YS II.50 tells us that we should try to produce a breath that is long (dīrgha) and smooth (sūkṣma). In other words, it's not just any normal breath.
To practise prāṇāyāma we need to chose the right seated posture, one which is comfortable and stable and one which allows us good and full access to our breathing for a reasonable period of time. There is no point choosing an āsana which will become uncomfortable and thus disturb our focus away from our Prāna (life-breath-energy).
Here is a list of five prāṇāyāma techniques I learnt from my teachers in the yoga tradition of TKV Desikachar and his father T Krishnamacharya that I would suggest we all ought to know about.
1 - Ujjāyī
This is a prāṇāyāma technique that requires us to breathe through both nostrils while gently constricting the muscles around the vocal chords. We will feel this technique at the back of the throat as a gentle rubbing sensation in the throat and for this reason it is sometimes known as Throat Breathing.
Ujjāyī produces a gentle sound a bit like the sound of the ocean when you hold a large seashell to your ear, and for this reason some call it the Ocean Breath.
Ujjāyī means something like "that which clears the throat and masters prāṇa". Sometimes ujjāyī is referred to as the Victorious Breath and that is probably a nod towards mastering prāṇa (used here to mean the prāṇa in area of the chest).
Ujjāyī is done by closing the mouth and inhaling through both nostrils while contracting the larynx so that is resonates gently, a bit like sighing, from the throat to the heart. It can also be done on the exhalation , especially when done during āsana practice.
Its effect is to lengthen the breath and bring our attention inward. It's a good one to get started with.
The effect of practising ujjāyī is heating and purifying.
Summary: Breathe in and out through both nostrils while making a gentle sighing sound in the throat.
2 - Anuloma Ujjāyī
Ujjāyī has many useful variations and one of them is anuloma ujjāyī. When we breathe in through both nostrils using the throat control technique described above and then close the right nostril and breathe out through the left nostril without the throat control this is called anuloma ujjāyī.
The HYP instructs us in this prāṇāyāma technique by saying that the yogi should close the mouth and gently draw in the air via the two nostrils so that it comes into contact with the area between the throat and the heart making a [sighing or soft hissing] sound and then exhale through the left nostril [HYP II.51 & 52]. Because these texts are often concise, we can infer that after breathing out of the left nostril we should breath in again through both nostrils using ujjājī and then out of the right nostril in the following breath. An easier way to remember this prāṇāyāma is to refer to it as "ujjājī inhale and alternate nostril exhale."
We control the flow of the out breath through each of the nostrils, first the left and then the right one, by using the mṛgi mudrā. The main picture shows someone using the mṛgi mudrā and the small picture inset in this section shows where to place the first and second fingers of the right hand, which the better hand to use for most people.
We use the right hand, if possible, to control the flow of the exhalation through one of the nostrils. The other hand rests on the left knee.
Breathe in using ujjāyī. At the end of the inhalation, gently place the thumb of the right hand onto your right nostril just below the bony part of the nose (not the opening of the nostril, but a little further up to where the cartilage of the nose begins) and the inside of your third finger gently onto your left nostril. Then breathe out through the left nostril slowly and smoothly.
The next breath follows the same basic instructions except that we breathe out through the right nostril by using the third finger (which gets a little support from the little finger) to close off the left nostril. The side of the thumb lightly rests on the left nostril.
The effect of practising anuloma ujjāyī is calming, which makes it ideal in our hectic modern world.
Summary: Breathe in through both nostrils making a gentle sighing sound in the throat, and out through the left nostril. Then, breathe in through both nostrils making a gentle sighing sound in the throat, and breathe out through the right nostril.
3 - Viloma Ujjāyī
If you have got the first two techniques straight in your head, then this one will be easier.
Instead of breathing in through the throat, so to speak, and out through one nostril, we do this variant of ujjāyī the other way around. This is called viloma ujjāyī and this is how you can do it:
Use the mṛgi mudrā described above to close off the left nostril and then inhale through the right nostril. You then exhale through both nostrils using the Ocean Breath (i.e. ujjājī). The next inhalation takes place through the left nostril and the exhalation takes place through both nostrils making that soft hissing sound we know as ujjājī.
The effect is stimulating and as such is often used these days in a modest way and or in conjunction with other techniques.
Summary: Breathe in through the right nostril and out through both nostrils making a soft sighing sound in the throat. Then, breathe in through the left nostril and out through both nostrils making a light hissing noise in the throat.
4 - Nāḍiśodhana
This method of prāṇāyāma is also called nāḍīśuddhi. Both refer to the cleansing of the nāḍī (the channels through which our prāṇa flows).
Again, if you have mastered the previous two techniques then this one should (I hope) be relatively easy to do.
Use the mṛgi mudrā as described above to control the flow of air through each nostril.
We usually start by breathing in through the left nostril (moon, iḍā nāḍī) and breathing out through the right (sun, piṅgalā nāḍī). The next inhalation starts in the right nostril and the exhalation takes place via the left nostril.
The effect of this technique is to balance our prāṇa.
Summary: Breathe in through the left nostril and out through the right nostril. Then, breathe in through the right nostril and out through the left nostril.
5 - Śītalī
This prāṇāyāma technique is done like this:
Start with the chin down and the tongue rolled up and fully extended (see picture)
During your inhalation lift the chin to just higher than the horizontal level
At the end of the inhalation, retract the tongue and curl it back and up towards the roof of the mouth such that the bottom of the tip of the tongue rests on the roof of the mouth.
Then breathe out through the left nostril
Breathe in again as set out in 1-3 above
Then breathe out through the right nostril.
There is a useful variation of śītalī whereby we can breathe out through both nostrils instead of alternate nostrils.
For those who can't fold, curl or roll the tongue into a straw like structure to do śītalī (and a significant minority of us can't), then there is sītkāṛi in which the tongue is placed in a different place. That is, the tongue is placed on the roof of the mouth, with the tip of the tongue touching the back of the lower front teeth and the lips are pursed. You may find that you make a quiet whistle when you breathe in. Well, never mind because sītkāṛi means the whistler.
The effect is cooling and calming.
Summary: breathe in through folded tongue and out through both nostrils or through alternate nostrils.
Summary of techniques
Prāṇāyāma Inhalation Exhalation
Ujjāyī 1. Throat Throat
Anuloma 1. Throat Left nostril
ujjāyī 2. Throat Right nostril
Viloma 1. Right nostril Throat
ujjāyī 2. Left nostril Throat
Nāḍiśodhana 1. Left nostril Right nostril
2. Right nostril Left nostril
Śītalī / Sītkāṛi 1. Tongue Both nostrils
 The Haṭhayoga texts don't agree on the details of the technique. For instance the descriptions in the Gheraṇḍa Saṁhitā V.69 and HYP II.51,53 are similar for the inhalation but not the exhalation.