Breathing is the most important fundamental of Tai Chi. Breathing is the foundation for Tai Chi!
Sadly, many masters do not know this. I have ran into Tai Chi masters that tell me that a person will automatically know how to breathe when doing Tai Chi. I have never seen this happen. A real master will instruct his students how to breathe correctly.
I have ran into students doing other martial arts especially from Taekwondo students that they were never taught when to inhale or exhale. Their kicks and punches are weak. They are injured easily. This is because improper breathing can cause Qi to block in the body and cause disharmony in the body. The disharmony is what makes these people injure easily. When breathing is not in sync with movements, you have improper flow of Qi.
Breathing can control the flow of Qi
Breathing is used to charge the body
Breathing creates power
Breathing can make the body relax and not tense
Breathing calms the Shen (spirit)
Breathing increases mental focus
GM Yang Jwing Ming often mentions the importance of breathing. He even teaches proper breathing with sounds to create energy manipulation in the body.