Connection between our breath and realizing Brahman or Self-realization is well documented. In fact, observing our breath leads to quicker spiritual realization than following other methods. This is explained in “Svara Yoga”, where svara refers to air breathed through nostrils. Breathing means both inhalation and exhalation. Swara Yoga is in the form of conversation between Shiva and His Consort Pārvatī. In that Shiva says, “स्वरः साक्षात्महेश्वरः॥svaraḥ sākṣātmaheśvaraḥ ||” This means breathing is nothing but Shiva. But, in order to realize Him through breathing, Shiva Himself revealed the procedure through Swara Yoga. This is also known as “इदं स्वरोदयं शास्त्रं idaṁ svarodayaṁ śāstraṁ”. This Scripture explains about our inhalation and exhalation. This happens through three main nāḍi-s (nāḍi is a tubular organ, like a hose pipe, through which prāṇa traverses). Prāṇa is different from air that we breathe; prāṇa is the vitality or vital force for the existence of the universe which exists in the air. It is the Cosmic Energy or energy of Prakṛti. Chāndogya Upaniṣhad highlights the importance of prāṇa in many verses. Nāḍi-s not only mean tubular vessels, but also subtle and yet powerful current passing from one place to another place; for example iḍa and piṅgala nāḍi-s. Prāṇa is the cause for the formation of foetus and the same prāṇa is the cause for the exit of the soul from the body. Thus, our whole existence depends upon on prāṇa only.
Swara Yoga explains how prāṇa can be utilized by us for all auspicious purposes. Iḍa nāḍi is the left nostril and piṅgala nāḍi is the right nostril. The third and most important one, yet highly subtle in nature through which prāṇa traverses is citriṇi nāḍi which is in the middle of suṣumna nāḍi or the spinal cord. When inhalation is made through our left nostril, the body cools down and that is why iḍa nāḍi is also known as Chandra nāḍi. When inhalation is made through piṅgala nāḍi, body becomes warm as it represents the sun nad is also known as sūrya nāḍi. These nāḍi-s are also called Śakti and Śiva nāḍi-s respectively. Ardhanārīśvara from of Śiva is based on this concept.
Svara Yoga
 
Suṣumna nāḍi is known as Brahma nāḍi (Brahman or Paramaśiva or Parabrahman). Inhalation through left nostril activates the right side of the brain (right cerebral hemisphere) and inhalation through the right nostril activates the left side of the brain (left cerebral hemisphere). Generally, svara shifts its activity from one nostril to another nostril approximately every 60 minutes. Activity of svara means both inhalation and exhalation. For example, during activity of left nostril, right nostril will be almost or partially blocked and both inhalations and exhalations will be only through left nostril; when inhalation and exhalation is through one particular nostril, that nostril is known as active nostril. At the time of change of svara from one nostril to another nostril, prāṇa enters suṣumna nāḍi for a brief moment and then becomes active in the other nostril. Though, it is said that one nostril is active for one hour, in reality, it varies from person to person and also depends upon prevailing weather conditions.

There are three main aspects in Svara Yoga:

1. Observing which svara is active.
2. Second, how to change the flow of svara depending upon the need.
3. Finally the benefits of prāṇa flowing through each of the nāḍi-s.
However, practicing yogic breathing and nāḍiśodhāna are prerequisites to practice swara yoga. They are explained in the article Basic Breathing techniques.

1. Observing which svara is active:

When breathing is done through left nostril, it means iḍa nāḍi is functioning and when breathing is done through right nostril then, piṅgala nāḍi is active and when breathing is equal through both the nostrils, then suṣumna nāḍi is active.
Active functioning of the nostrils begins 90 minutes before sunrise every day. Sunday, Tuesday and Saturdays it begins with right nostril and on Monday, Wednesday, Thursday and Fridays it begins with left nostril. This is related to planetary influences (navagraha) on our breath. Out of this 90 minutes, only during the first 60 minutes, nostrils become active as above. During the next 30 minutes (out of 90 minutes) nostril of the day takes over and this is known as process of transmission.
Active svara depends upon śuklapakṣa (the brighter half, of a month, the 15 days of the moon's increase or waxing moon or from amāvāsya to pūrṇamāsa) and kṛṣṇapakṣa (the dark half of a month, fifteen days during which the moon is on the wane, time from full to new moon or the waning moon or from pūrṇamāsa to amāvāsya). During the first three days of śuklapakṣa, left nostril (iḍa nāḍi) is active during 30 minutes (last 30 minutes of 90 minutes mentioned above) and during the first three days of kṛṣṇapakṣa right nostril (piṅgala nāḍi) is active during those 30 minutes. Let us assume that sun rises at 6.00 on a Sunday. 90 minutes before the sunrise is 4.30 early morning. From 4.30 to 5.30, right nostril (piṅgala nāḍi) is active. Suppose if it is Monday then, left nostril (iḍa nāḍi) is active. After this, during the first three days of śuklapakṣa, left nostril is active, next three days right nostril is active and again the next three days left nostril is active. Thus during waxing moon (śuklapakṣa) 1, 2, 3, 7, 8, 9, 13, 14 and 15 tithi-s (tithi means lunar days; from first day of waxing moon to the last day of waning moon, it is 29.531 days; this is known as lunation or lunar month), left nostril is active. During waning moon (kṛṣṇapakṣa) 4, 5, 6, 10, 11 and12 tithi-s, right nostril is active. The first svara of the day is known as svara of the day.
This can be explained this way. We are taking 4th day of śuklapakṣa, which is Friday as example. Sun rises on this day at 6.00. On this early morning between 4.30 and 5.30, left nostril will be active (because it is Friday). 5.30 to 6.00 will be change over period from nostril of the week day to the svara of the day (lunar day based on tithi). Since this is this 4th day of śuklapakṣa, right nostril will be the svara of the day. During 30 minutes, there will be transition between svara of the week day and svara of lunar day. From 6.00 am on this day, right nostril will be active. After that approximately every one hour, svara will alternate between the two nostrils. There will be some duration adjustments to match with the svara of the next day. Since next day is śuklapakṣa Saturday svara of the week day (4.30 to 5.30) will be right nostril and svara of the day will be right nostril too. In case the above Friday falls on 4th day kṛṣṇapakṣa, the svara will be like this. 4.30 to 5.30 left nostril; 5.30 to 6.00 will be the transition period and from 6.00 to 7.00 left nostril will be active. Thus, a svara can be active more than 60 minutes too.  Thus it can be observed that once in three days, svara of the day changes at the time of sunrise. When svara is active in the left nostril, right brain is active and when svara is active in the right nostril, left brain is active. It is always important that force of the exhalation should be reduced to ensure that it takes long time to exhale. Prāṇāyāma rule says that duration of exhalation should be much longer than inhalation, which is normally twice the duration of inhalation or more. 
Practically, it may not be possible to calculate swara all through the day by observing active svara at the time of sunrise. Simple way to know the active svara is forcefully exhaling into right hand index finger, by keeping it very close to nostrils as shown in this image.
Svara Yoga
 

We can easily observe which nostril is active from the force of the breath from each of the nostrils. Breaching through each of the nostrils have certain benefits. According to the need at that point of time, svara has to be activated to a particular nostril. May be that svara is active during that time and in this case, we need not do anything. In case, the svara is active on the other nostril, then we have to change the svara to the required nostril. Duration of svara in a nostril need not be exactly 60 minutes. It could wildly vary. Therefore, it is not advisable to go only with prāṇa chart. Svara also depends upon one’s inherent guṇa-s (sattva, rajas and tamas) and dhātu-s (vāta, pitta and kapha).  Guṇa-s can be observed from one’s behaviour and dhātu-s can be observed from one’s pulse. These information are only supplementary in nature and will be required when we try to activate a particular nostril or svara. How to read one’s pulse is explained in this external link

Ayurveda pulse Diagnosis

2. How to change the flow of svara depending upon the need:

By practice, which is called sādhana, svara can be made to pass through suṣumna. When svara passes through suṣumna, it keeps the mind calm and composed, which ultimately leads to realization. This has been dealt with elaborately in Theory and Practice of Kundalini Meditation.
If we want to activate our left nostril, the best method would be to close the right nostril with a finger. Both inhalations and exhalations should be through left nostril only. Ten round of breathing will normally change the svara. While lying on the bed, if we turn in such a way that our desired nostril is up and the active nostril is down, then the desired nostril will be active. Alternatively, when active nostril’s side armpit is pressed (ancient sages and saints used daṇḍa for this purpose), the other nostril will become active. There is yet another way to activate the desired nostril by looking at the desired nostril using both the eye balls. This will take some time as it requires practice.

3. Benefits of prāṇa flowing through each of the nāḍi-s:

When left nostril or iḍa nāḍi is active, following auspicious things will happen. All auspicious things can be commenced during active left nostril. One can interact with one’s spiritual Guru by activating left nostril. But most of the times, if a disciple is sitting before his Guru, both the nostrils will become active. However, this depends upon the energy level of the Master. If iḍa nāḍi is activated, all actions can be completed with successes and with ease. Iḍa nāḍi can be activated irrespective of whether it is day or night.
Right nostril need not be unnecessarily activated, as most inauspicious events are attributed to piṅgala nāḍi. Tantra teachings are commenced during active right nostril. When right nostril is activated during eating, digestion will be good. Gratification of senses also happen during active right nostril.
Suṣumna is active when spiritual energy is very high. In fact, flow of spiritual energy happens only when suṣumna is active. This is called śūnya svara. Suṣumna represents fire, as against iḍa nāḍi representing moon and piṅgala nāḍi representing sun. No auspicious things should be performed when suṣumna is active. It must always be understood that material life alone can lead to spiritual life and not vice versa. Both iḍa and piṅgala represent material world and suṣumna spiritual life. Suṣumna can be activated during meditation and all spiritual related activities. Well activated suṣumna will trigger trance (samādhi).
Therefore, by controlling our breathing pattern and by properly practicing prāṇāyāma, one can be highly successful both in spiritual and material lives. Perfect spiritual path can be pursued only if one has proper material life. Relinquishing material life is not the right way to realize Brahman. By remaining in the material world, one has to properly seek spiritual path. Swara yoga will be of great help in achieving perfect balance between material and spiritual life.
Further Readings:
Breathing Exercise
Stretching Exercise

Breathing and Exercises