Circular breathing is a technique used by wind musicians1 which allows them to play long passages without a break. Essentially, it means breathing out as normal then, just before you run out of breath, storing some air in your cheeks. You then use your cheeks to squeeze this air out of your mouth, while at the same time breathing in through your nose. You then have a fresh lungful of air to breathe out.
So, how do I do it?
Well, it takes practice. It helps if you start off with the basic principle of squeezing something out of your cheeks while breathing in through your nose, then build up from that. This is possibly the best method of learning:
The most difficult thing about circular breathing is the main concept - being able to breathe in through your nose while expelling air through your mouth. Normally, you wouldn't need to be able to do such a thing, so it seems impossible at first. It creates mental confusion, like trying to rub your stomach while patting your head. The first step towards overcoming this is very simple; take a mouthful of water, and lean over the sink or a bucket. Open your mouth a little way, and let the water drain out slowly. While the water drains from your mouth, breathe in and out through your nose.
This seems like a small progression, but it can take a lot of effort. It is essentially the same as step one, except you now push the water out of your mouth, rather than just letting it drain out. To do this, you should use your cheeks and tongue - not air pressure from your lungs. Once you've got used to pushing the water out, start breathing through your nose again. This is the basis of circular breathing. You now have to transfer this skill to pushing air, rather than water, out of your mouth.
To be sure you are pushing air out of your mouth, you need to be able to see it. One of the best ways to achieve this is to use a straw in a glass of water. Remember, you should only be using your cheeks and tongue to push the water out.
After a little practice, you should breathe in through your nose at the same time as doing step three. Take as big a breath as you can, while still pushing air out from your cheeks.
Now you have to put it all together, breathing out normally through your mouth, then just before the air runs out, storing some in your cheeks. You then do step four, taking as large a breath as you can. Then you can start breathing out normally again... you are now circular breathing. You should watch the bubbles in the water, and make the stream as constant as you can, especially when changing from lungs to cheeks: this will make a constant airflow in the instrument, giving a constant sound.
And that's all there is to it.