The nose is for breathing, the mouth is for eating.

Breathing.  We do it all day, everyday, involuntarily and most of the time we have no awareness that it's happening.  You can go for about a week with out food, a few days with out water, but without your breath?  To quote Ana Forrest, "you're dead meat in TEN MINUTES".
When we breathe properly, in and out of the nose, taking in as much air as possible, some amazing things happen!  First of all, breathing through the nose filters and cleans the air of most of the impurities before it enters our bodies.  (Nose hairs have a purpose!)
Try this: Take a few breaths in and out through your mouth.  Feel how the air moving over your tongue feels a bit cold?  When we breath through the nose the air is in the body for just a bit longer and therefore the air is warm.  When we are breathing in warm air we increase our metabolism and burn free radicals.  Just from breathing!  Cool, right?
And as for breathing deeply, well, most of you probably know that as adult humans we rarely use our lungs to their full capacity.  When we don't fully inhale and exhale completely air can settle into the bottom of our lungs and become stale.  Kinda gross?  So by taking in air fully and expelling it completely we are insuring the health of our lungs and our entire body.  Our cells need oxygen as a source of energy to perform the functions for which they are designed.  The more oxygen we take in, the more for our cells to use for energy!
Our breath is our life force.  And even though we don't have to be aware that we are breathing to stay alive, when we are aware that we are breathing we can fully LIVE.  Bringing awareness to the breath brings you immediately into the present moment.  Your breath is not caught up in what is was just doing, what it's about to do.  It goes with the flow.  In and out.  It's going to happen whether you observe it or not, but when you observe your breath, your mind starts to clear.  The chatter inside your head gets quiet if only for a moment, one breath.  BUT the more you observe the longer the stillness in the mind can stay with you.  The mind is wild, but when you can control the breath you can begin to control the fluctuations of the mind.
Pranayama is a branch of yoga focused on breathing techniques to create different sensations in the body.  Prana is translated to "life force" and yama is "to control".
If you are feeling sluggish and want to increase energy in your system close off your left nostril and breathe in through the right nostril only.  Then block the right nostril and breathe out of the left side.  Close your eyes and continue breathing like this for a few minutes and see how you feel!
If you want to wind down, if you're having trouble sleeping or just want to relax, try the opposite.  Block the right nostril and breathe in through the left then to exhale block the left and let the air out through the right.
After trying either of these two exercises, be sure to take a few normal breaths through the nose, to take the time to see how your body feels after trying and to bring you back to balance.

There are many ways to control the breath to feel different things, this is just a simple one.  I hope you try it, let me know how it goes!  And always when you breathe, give thanks and don't take your breath for granted.