The Importance of Balance
Let's ignore the more philosophical elements of the term "balance" for the moment. Balance is a basic principle of motion. Walking, for example, has been described as a controlled loss of balance that causes us to move forward.
 
Balance comes in two forms: static, i.e., standing still; and dynamic, i.e., moving. Practicing karate has been shown to improve both kinds of balance through the repetition of the proper karate actions. These movements are primarily designed for more effective delivery of karate techniques, but have the side benefit of improving this useful aspect of movement in general.
 
Balance and stability are necessary to basic techniques. When one leg supports the entire body during a kick, or when shifting either your own or your opponent's body's center of gravity for a deflection or throw are examples of techniques that require an understanding of balance.
 
Controlling your opponent's body is called "Happo No Kuzushi" in Japanese. "Happo" literally means eight (or all) directions. "Kuzushi" means "to destroy or break down". Understanding this principle allows you to destroy an opponent's balance in eight, or any, direction.
 
"Tai sabaki" is the technique of controlling your own body movements to evade, deflect or counter an attack, as well as what Aikido practioners call "blending your force with your opponent's", which is largely a function of joining your center of gravity with your opponent's and leading their body to where any attacking force is nullified.
 
Of course, it's necessary for a beginning student to practice seemingly endless hours of drill to achieve good balance. There are two simple points that, once mastered, will allow the karateka to move on to more involved techniques with confidence.
 
The first principle is to have a solid foundation from which to act. This may seem obvious, but lifting a heel or turning your body at the wrong moment can rob you of a very large portion of your effectiveness, not to mention exposing you to your opponent's techniques.
 
For almost all techniques, one leg or the other bears more weight than the other. This is your supporting leg, and should have three points of contact with the floor - your heel and the ball of your foot near the little and big toes. The toes themselves should not be used as supporting elements, except perhaps where a technique is meant to be holding something down. The knee of the supporting leg should be bent and the knee vertically aligned with the supporting foot. The degree of bend will depend on the technique in use. Keeping your supporting foot flat rather than letting it roll onto it's edge will cause your hip to align itself as well. If you are standing on one leg, your whole center of gravity should be aligned over the supporting foot and the springy bent knee. A bent knee gives you the ability to shift and move your body as necessary, which is one of the reasons that your leg is rarely fully extended in karate.
 
Since most of us don't spend a lot of time statically standing on one leg, this may appear to have limited use. However, even with both feet on the ground, we dynamically shift our weight (and center of gravity) from leg to leg. The same principles apply. A savvy opponent can time our footsteps and sweep our legs out from under us when we are momentarily unbalanced, so it's necessary to try not to commit to a given stance until the last possible moment.
A stance can be narrow or wide - each has it's uses. A narrow stance decreases stability, but increases mobility. The opposite is of course true for a wide stance. This difference in purpose is a hallmark of the differences between styles like Goju-ryu, Shotokan and Shorin-ryu.
 
The second principle involves keeping your eyes (and therefore your ears) level. Tiny hairs in the ear canal are used to sense your orientation in space. You sacrifice the ability to maintain your balance when you tilt your head. How many times have we been told to watch our opponent, keep our eyes up, etc.? If our opponent is the same size as we are, as it is when performing kata, then we will naturally be eye-to-eye with them, and therefore level.
 
Of course, it's not always possible to stand still, or even remain vertical. Two useful tricks to gaining a sense of balance are to practice visualization of your own and your opponent's center of gravity/vertical body line, and to train in slow motion, such as is performed in Tai Chi, for example. A "slow-mo" kick has the added advantage of gradually increasing muscle strength and flexibility. Enough reading! Time to train!