For the better part of history, astronomers have known that the Sun is the center of the universe, with the Earth, stars and planets all making their revolutions around it. In terms of the small part we play in this vast universe, it serves as our center, as our core.
Similarly, our personal core could be considered the center of the human body. By most definitions, the core refers to the torso, (minus the legs and arms) including the belly, the hips, shoulders, neck, mid and lower back.
Just like the Sun is the center of the Universe, the Core can then be considered the center of a Boxers Fighting Range, which is basically his universe. Because all functional movements are highly dependent on the core, every movement that emanates from it is directly affected by it. Your boxing universe orbits, along with all of its parts (every punch you throw and step you take) around your core, so your center should never venture outside of it.
An important trait for you to master as a fighter is to be sure to keep your core constantly at center. You should be shifting your weight forward as you punch, side to side as you alternate punches, and in perpetual motion as you act and react, but your weight should always return to the center.
For one, when you punch, you should never extend your head past your front foot. It should never go beyond the point of what can be foundationally supported or outside of your Fighting Range. When it does, your stability is compromised and you take yourself out of the proper position to act or react. When you punch and leave all of your weight on your front foot, you're out of balance with your universe.
You see fighters do it all of the time. They jab with all of their weight over their front leg or reach in towards their opponent, without having their feet beneath them. This takes away most of the power a fighter possesses in his right hand (left cross if you are a southpaw) and, even though it gets a fighter's left jab closer to his opponent, it makes his jab less effective, from a power perspective, than it should be.
It's not only important to keep your weight centered from an offensive standpoint, but from a defensive one too. If the majority of your weight is placed on your front foot, how do you slip a right hand? Try it. It simply doesn't work.
This applies to your back foot as well. Any time, in relation to any moving in any direction, when you let your weight drift outside of your designated Fighting Range you are going to compromise yourself. In terms of punching power, balance, your ability to move, defend properly and set yourself up to deliver the next shot, keeping your core at the center is vital.
Having your weight equally balanced and distributed between your feet, some bend in your knees and using the shifting of your weight to generate power and momentum will make you feel and react like a more solid fighter. You will be better prepared for anything that comes your way and ready to move heaven and earth. From the core of your being, you will have the foundation to be a better fighter and will hold the universe in the palm of your hand.