The barycenter is the center of mass of two or more bodies which are orbiting each other, and is the point around which both of them orbit.
In the case where one of the two objects is significantly larger and more massive than the other, the barycenter will be located deep within the larger object and rather than appearing to orbit it will simply be seen to "wobble" slightly. This is the case for the Moon and Earth, where the barycenter is located about 4,700 km from Earth's center and well within the planet's crust. When the two bodies are of similar masses, however, the barycenter will be located outside of either of them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, Jupiter and the Sun, and many binary asteroids and binary stars.
The distance from the center of a body to a barycenter for a simple two body case can be calculated as follows: