For following explanation, by breathing, I mean, inhaling and exhaling through nose or nostrils, and not through mouth.
In Sanskrit language, the practice is called Pranayam
It is made up of two words, Pran(a) and Ayam(a). Pran means spirit, soul, and ayam means something like exercise or such practice.
Put simply, it is exercise of breathing to increase capacity and power of lungs, feel at peace, relax, etc. It is done in four stages as below.
1. Inhale : Breathe in slowly.
2. Hold your breath : Neither inhale nor exhale.
3. Exhale : Release your breath slowly.
4. Hold your breath : Neither inhale nor exhale.
The four stages are carried out in some proportion as @augustlan said. I don’t remember. Ah, you may refer this site : http://www.feelingsoulgood.com/index.php?id=5
In addition to physical benefits, there are many other benefits too, like mental benefits (feeling calm).
Swimmers or deep water divers must be trying breathing control.
It can also be used as a tool for concentration. Notice, focus or experience the air moving in and out of nostrils, touching skin inside, hair in nose, etc. After some time, you will be good deal concentrating.
Edit : in this exercise, we are supposed to take in air in lungs, so that lungs rise. Doing this may qualify it to call as exercise.
Breathing naturally or unconsciously, air is taken in and stomach rises, as you may notice while others are sleeping (sleeping dogs, for example). It is used by body for relaxing.