I agree with everything that Amy Matthews describes above. In terms of what to do to enhance breathing I have 3 quick tips. Each is adapted from a great teacher. They are:
1. When in doubt let breath out.
2. Change your rhythm to change your breath.
3. Calisthenetics for the breath can also be good.
Tip #1. - This tip is an adaptation of an adage from my wise father who always said - when in doubt throw it out. When I am teaching group classes or working with individual clients I find myself advising them to exhale by saying “When in doubt let the breath out.” As Amy mentioned there are numerous reasons that people hold their breath. To relax this tension and to get breathing more fully it is helpful to let the stale air out of the lungs. so tell yourself - When in doubt blow it out! With the lungs empty the natural instinct to bring in fresh oxygen will kick in. Sometimes folks get stymied by all the different instructions about breathing, this is a reminder to simply start with an exhale!
Tip #2. Change your rhythm to change your breath. I once asked one of my favorite teachers of Bartenieff Fundamentals, Fran Parker, what Irmgard Bartenieff and Rudolf Laban advised about enhancing breath. Fran shared that it was her understanding that Laban instructed students to simply change what they were doing in order to improve breathing. Since breathing is an automatic process if one changes ones pace, rhythm, or task the breath will automatically adjust. So if we want to breath more deeply we can move more vigorously. If want to breath more calmly we can move more calmly. By changing the dynamics of your movement you are stretching your breathing apparatus to new dimensions.
Tip #3. Calisthenics for the Breath. Speaking of calm, yoga is a wonderful calming practice, a great teacher. Yoga also has numerous exercises for the breath, pranayama. These are like calisthenics for the breath, strengthening the pathways to breathing more deeply. Sometimes its is great to consciously do movement that brings strength and awareness to the parts of the body involved with breath - nasal passages, throat, rib cage, and diaphragm. Breath of Fire and Alternate Nostril Breathing are two examples of practices that prepare the body for clear and easy breathing. From a somatic perspective, Bartenieff also taught us to bring conscious awareness to the 3dimensionality of the diaphragm and its impact on the entire torso. When we are aware that the diaphragm inserts all around the lower ribs to the back of the spine we begin to breath deeply into our lower back as well as our abdomen. This is a great massage for tight back muscles. It is often best to feel this when in a hunched over position (this keeps us from only breathing in the belly and allows the breath movement to occur toward the sides and back).
So, when in doubt, breath out and enjoy your full dimensionality.