Circular breathing is an advanced playing technique used by saxophonists and other wind instrument players.
When performed properly, the skill allows players to produce a continuous tone without requiring them to stop for breaths. A player using this breathing technique must blow out through the mouth while breathing in through the nose. While this sounds impossible at first, the secret is storing extra air in the cheeks and using this air to continue the tone while breathing in.
A player must first take a full breath and begin to exhale. Near the end of the breath, the player blows the last bit of air from the lungs to the cheeks, inflating them and holding the air.
As the player compresses the cheeks and blows out the final volume of air, he or she must inhale through the nose very quickly. The player must take in as much air as possible before the extra air in the cheeks is gone.
While this is difficult to do and even more difficult to do without obvious shifts in tone quality, players that can circular breathe can play for huge stretches of time without stopping to take breaths. In fact, saxophonist Geovanny Escalante used the breathing technique to set the world record for continuous playing for just over 1 hour, 30 minutes. Previous to this, Kenny G held the record for holding a note on his Soprano for 45mins.
You will soon find more specific instructions on how best to learn and incorporate this into your playing here.
While most Western music does not require players to use any form of cyclic breathing techniques becuase the length of phrases are not overly long, many saxophonists still find the technique extremely useful. It allows for pieces that are normally only performable on string instruments to be transcribed for saxophones and other wind instruments. The other reason you may be looking to learn this technique is to play very long solos, all in one breath.