Learning how to breathe correctly is a very individual process - what works for one runner might be useless for another. But the benefits of learning to breathe properly (greater flow of oxygen to your muscles and hence greater endurance) are certainly worth some investigation, and there are certain techniques and guidelines that should be useful in this process.
First and foremost, you have to learn to locate your breathing correctly. Many runners start out breathing from their upper chests - you can tell because they look like they're heaving, especially at high speeds. It is much better for your stamina to breathe through you diaphragm, or lower abdominal muscles. You should try this first when you are walking, or even sitting, since you can practice relocating your breathing without distractions. Push your belly out slowly and feel your lungs fill up, and then slowly release. Try this for a few minutes to get the feel of it, and then try transferring it to your runs.
There are disagreements as to basic breathing technique (some people advise breathing in through your nose and out through your mouth while others suggest other patterns) so you should probably try some different combinations (in through your nose/out through your mouth, in and out through your mouth, in and out through your nose) and find what works best for you.
You will want to learn to coordinate your breathing and your stride, as it is very helpful for your endurance. Start out breathing in for 3-4 steps and then out for 3-4 steps, keeping your stride and breathing as even as possible. If you have difficulties keeping this pattern you might be running too quickly - you should be able to speak complete sentences as you run (particularly if you are a beginner), and if you aren't able to it would be better to slow down. If you are having problems maintaining the rhythm, try developing it at a walk and then moving on to a run.
After you have developed this basic technique, you can shift the proportion of your breathing somewhat. Some coaches advise a 3:2 ratio of breathing in and breathing out (enabling you to breathe in more oxygen in a given time period), though when you are running quickly it can turn to a 2:1 ratio. As you advance as a runner you might also want to do some core exercises to build up your diaphragm, which will allow you to control your breathing even further.
So overall, give these techniques a try, and remember to stay aware of your own body's particular requirements.

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