The most famous breathing method is Tanden breathing method (Tanden Kokyuho).
It is said that Tanden breathing method (Tanden Kokyuho) is the Lord Buddha (Siddhartha Gotama)'s breathing method. I have never met him in person and it is impossible to ask him now if it's true or not, so I'm not sure. But anyway, sure it may be an admirable breathing method because it is said that the Load Buddha (Siddhartha Gotama) practiced this method.
Whosever it is, I would like to explain briefly about this Tanden breathing method (Tanden Kokyuho) here.
Step 1. Sit on the chair. Sit up straight but not too much, be natural. And focus your attention on Tanden.
Step 2. Put your fingertips on both of your hands on the pit of the stomach, press gently, and as bend your body sloooooowly, breathe out through your nose sloooooowly till empty all air, feeling as though exhale air from Tanden. No need to exhale too much like you faint dead away. Take it easy.
Step 3. When you empty all air, as you slowly return your (upper) body to an upright position, loosen the press on the pit of the stomach gradually, inhale air geeeeeently through your nose into Tanden. Your (upper) body back to the original position, inhale fully untill you lean back a little bit. This time too, take it easy. You don't have to inhale too much like you feel pain.
Step 4. When you inhale completely, again, press the pit of the stomach gently, and as bend your body sloooooowly, breathe out through your nose sloooooowly till empty all air, feeling as though exhale air from Tanden. Don't do too much, take it easy.
Step 5. Empty all air and your body bent, back to Step 3.
Repeat the procedure above as much as you want.
And that's all.
There are various ways depending on each teacher/master/leader/instructor, however all of those are built upon the same foundation; inhale air geeeeeently through into Tanden and breathe out sloooooowly till empty all air, feeling as though exhale air from Tanden.
Some say that you should never hold your breath while you do it, some say when you inhale completely, hold your breath three seconds and release, some say it have to be Sanko Ikkyu; exhale three times which is "one short, one short, one long" like "ha ha haaaaa", and inhale one time. Some says it should be Harosoku, some Daishinsoku, some this some that and some something else. But after all, it all comes down to "inhaling deeply and exhaling completely focusing attention on Tanden".
Some say you should breath out with a hum "nnn". I guess this idea comes from a yoga technique of breathing control called Bhramari Pranayama. Anyhow, as Tanden breathing method (Tanden Kokyuho) itself is said to be the Lord Buddha (Siddhartha Gotama)'s breathing method, sure the root of Tanden breathing method (Tanden Kokyuho) can be Yoga. So you are free to hum or not, as you like.