The center is located halfway between the navel and the waistline, and roughly halfway inside of you. The Japanese call this point the hara; the Chinese call it the tan-t’ien. According to Traditional Chinese Medicine, the center is a “lake” of ki, which feeds life-energy to the rest of the body through a series of tributaries ("meridians"), just like how a car's master cylinder compresses brake fluid. Likewise, since the center is said to be the exact physical location of the immortal soul, it is one of the body’s seven chakra, and several energy vortices. In Goshin-Jutsu, we ignore all of these things -- not because they have no verifiable basis in reality -- but because they have no practical application if they did.
We use the term “center” to help divorce ourselves from mysticism. The center is merely the location of the center-of-mass on normally-proportioned people. From a physics standpoint, people can be treated as a single dot, floating in space, which weighs as much as a person. As a teaching tool, this is visually represented on normally-proportioned karateka by the knot of their belt. When the knot of your belt is below your opponent’s knot -- when your center is lower than your opponent's center -- you have leverage, and the resulting mechanical advantage will multiply your strength.
[picture of a karateka being reduced to a dot]