Having core stability means that your core muscles are strong and able to support the movements of your body as you use your other muscles to move yourself. It helps you keep your balance, helps to avoid injury from sudden or new movements, and is responsible for good posture.
One of the major components of developing core stability is to breathe properly using diaphragmatic breathing. Unfortunately many of us have not learned to breathe properly which leads to many imbalances in the body. Once you're breathing properly you can move on to specific core stability exercises.
Although we do not get into these complexities or have the same goals and methods, this video is good for discovering and developing an awareness and relationship between breathing and your body. This helps to develop the mind-body connection which will help you in all aspects of your life, including your fitness.
Diaphragmatic Breathing and Core Function
Susan McLaughlin, PT describes the relationship of the diaphragm and the pelvic floor for core function. Restoring movement of the diaphragm and pelvic floor is crucial for dynamic stability of the core.