Babies instinctively know how to breathe deeply, but develop bad habits that promote shallow breathing by the time they get to school.
Breathing seems easy enough for healthy people. We do it all the time without thinking about it. Unfortunately, that can be a problem. While babies instinctively how to breathe properly, by the time they start school they have usually developed poor posture and other bad habits that lead to shallow breathing. As they grow older, people maintain these habits into adulthood unless they have some reason to learn otherwise. There are even times when we forget to breathe at all.
I remember when my second grade second grade teacher had us out for recess, had us get in a circle and try to see who could stand the straightest. Even at that age, I thought it was funny watching people turn all rigid with their shoulders up around their ears, although mine probably were, too. Were any of us breathing? I doubt it.
When I was teaching trombone to beginners, I noticed that if I asked a student to take a deep breath, the chest went out and very often the shoulders went up to the ears. That is not the way to take a deep breath. Most schools begin their band programs in fourth or fifth grade, and while I have had little recent contact with children that age, I have the impression that by that age no one breathes as naturally as they did as babies. Watch a baby. If you don't have a baby around, a dog or cat can probably teach the same lesson. As babies breathe, their shoulders do not move. Their ribcage moves only a little if at all. Their abdomens expand and contract as they inhale and exhale.
Singers and players of wind instruments are taught to breathe, and I suppose so are dancers, athletes, and others. It's also an important part of yoga. Older children and adults can easily relearn to breathe deeply with time and attention, whether they study anything like singing that demands attention to it or not.
Sit or stand with your back straight, your arms and shoulders relaxed. Place both hands on your waist, with your fingers by your abdomen and your thumbs by your back. Round your mouth as if you were going to say, "Oh." Thinking of that sound, expand your abdomen. (This is one of those times that, if you're thinking about your hands or your abdominal muscles, you are likely not to breathe at all!) You should feel the expansion of your abdomen all the way around. As you expand and contract your abdomen, you are automatically breathing deeply.
Pay careful attention to all of the physical sensations: the motion of your abdominal muscles, the sensation of the air rushing in and out of your mouth. (If you really thought about saying that vowel sound, you're breathing through your mouth.) Close your mouth, and continue breathing through your nose. Keep concentrating on the physical sensations, remaining mindful for several minutes. Repeat the exercise frequently enough that deep breathing becomes a habit.
There are times when attention to breathing becomes critically important. Singers and wind instrumentalists (breathing through their mouths) must be able to inhale more quickly and then exhale more slowly than regular breathing. Learn the quick inhale by snapping your abdomen out. Then exhale over a count of, say, ten seconds. Musicians, at least, should also notice that getting the maximum amount of air involves expanding both the abdomen and the chest, and that it is possible to expand them together or separately in either order. Practice mindfulness of the sensations of these movements separately from using them in a musical situation until you get the hang of it. Then practice playing or singing real music--simple and familiar music at first, working up to more difficult music.
Careful attention to breath is also important for lifting weights, whether as part of an exercise routine or just picking up something heavy to move it. As part of an exercise routine, it doesn't matter whether you are working on an exercise machine, using free weights, or just moving part of your own body. Failure to breathe properly can be dangerous. It's one of those times when it's easy to forget to breathe entirely.
Before starting an exercise or other activity that involves weights, be mindful of your breathing for a few breaths. Once you start, exhale as you pull or push on the weight (that is, while your muscles are contracting). Inhale as you return to your starting position. Especially if you are doing an exercise routine, careful attention to breathing is every bit as important as careful attention to any other part of your technique.
Deep breathing is also excellent for stress relief, at least partly because we can't really concentrate on two things at once. Mindfulness of breathing takes attention away from whatever thoughts are causing the stress, and that is helpful above and beyond the physical benefits of good breathing.