James Nestor, on pp. 41-42 of his book Breath: the New Science of a Lost Art, claims the following.
The right nostril is a gas pedal. When you're inhaling primarily through this channel, circulation speeds up, your body gets hotter, and cortisol levels, blood pressue, and heart rate all increase. This happens because breathing through the right side of the nose activates the sympathetic nervous system, the "fight or flight" mechanism that puts the body in a more elevated state of alertness and readiness. Breathing through the right nostril will also feed more blood to the opposite hemisphere of the brain, specificially to the prefrontal cortex, which has been associated with logical decisions, language, and computing.
Inhaling through the left nostril has the opposite effect: it works as a kind of brake system to the right nostril's accelerator. The left nostril is more deeply connected to the parasympathetic nervous system, the rest-and-relax side that lowers temperature and blood pressue, cools the body, and reduces anxiety.
... Our bodies operate most efficiently in a state of balance, pivoting between action and relaxation, daydreaming and reasoned thought. This balance is influenced by the nasal cycle, and may even be controlled by it. It's a balance that can also be gamed.
A question on this site describes we only breathe out of one nostril at a time: at any given time, one of our nasal cavities will be more open than the other, and this switches back and forth every couple hours. For the most part, we are unaware of this. (At any rate, I was!)
I am very curious about Nestor's claim that the balance can be gamed. It seems that one could prepare effectively for a math exam by practicing right-nostril breathing, or for sleep by practicing left-nostril breathing. Is this the case?