The left nostril (or the flow of Prana or vital energy in the left channel) is represented by the Yamuna, the river associated with the deity Krishna, who represents aspects of Bhakti Yoga or Yoga of Devotion (surrender, love towards the Supreme, service, etc.). And these emotions are heightened and refined by focusing on the life energy in the left nostril. Yamuna is also the sister of Yama, the deity of death and hence Yamuna is said to free one from the fear of death.
Ganga on the other hand represents the path of Knowledge or Jnana Yoga (non-duality, self-inquiry, contemplative practices, etc.) - regulated by the right channel, accessed through the breath in the right nostril.
There is a System called the Svara Yoga where one can consciously switch the breath to the desired channel at specific parts of the day (think of it on the lines of TCM where certain aspects of Yin/Yang or the elements influence different parts of the body at different times in the day). Some use this branch of Yoga to gain intuition and awaken psychic faculties, other to prolong life and others to attain stillness. Until perfect stillness is achieved, the breath does not become subtle enough to enter the central channel or Sushumna, the road to Samadhi (absorptive state of bliss where non-duality is glimpsed).
If you notice carefully, breath pre-dominantly flows through one of the two nostrils at any given point in time :) This would be the dominant channel at that moment. This is also used in Ayurveda (traditional Indian medicine) for diagnosis.
While the Indian version of Svara Yoga is rudimentary, the Tibetans, in their system called the Trul Khor, go to great depths in manipulating and refining the two currents to enable its switching into the central channel naturally (by employing physical postures, force, mantras, mandalas, visualization, breathing techniques and sexual alchemy at the advanced stages).