In common with a lot of the Sanskrit terms used in Yoga, Pranayama has an ordinairy meaning, a deeper meaning and a number shades of further meanings.
It's built up from two words Prana and Ayama.
The surface meaning is, for prana - breath - and for ayama, which is the opposite of Yama (restraint), unrestrained. So Pranayama means breath without restraint.
The deeper meaning is, for prana - life - and, for ayama - not death, so here it means life (without death).
Pranayama also means the different ways in which we practise breathing in order to achieve unrestrained breath and long, if not eternal, life.
In his very, very good book 'religiousness in yoga', Desikachar says the pranayama can be (in the beginning) the simple act of becoming aware of breathing. He then goes on to describe some of the ways, techniques and practices we can apply in our daily life to deepen, stretch and strengthen both our breath and our powers of concentration.
There are additional benefits. Deep steady breathing has a calming affect on both body and mind, so it's useful for combatting stress. For people who don't find much time in their life for the kind of exercise that get's the heart-rate up and the lungs relly working, it can be quite transforming to start breathing in a way that gets more oxygen into the blood stream