The barycenter.

The barycenter of the Earth - Moon system is inside the Earth about 2000 kilometers beneath the surface.

The Barycenter of the Charon - Pluto system is in space between those celestial bodies.

In an ellipse, as described by Kepler's laws. Though this is usually applied to the planet, it is more accurate to apply it to the barycenter of planet + moons.

The "barycenter" of any system - for example, the Earth and the Moon - is the center of mass, which is also the center of the orbital motion within the system. For Pluto and its "moon" Charon, the "barycenter" is somewhere between the two objects, which are of (very roughly) similar size.

In the Earth-Moon system, the "barycenter" is not the center of the Earth itself, but is still located within the Earth. If it were not, then the Moon wouldn't be a "moon" at all; it would be a co-orbiting planet. So technically, Charon isn't a moon of Pluto, but a co-planet.

Because the Sun contains about 99.5% of the mass of the entire solar system, the barycenter of the solar system is very close to the center of the Sun. Not quite exactly; Jupiter's mass moves the barycenter of the system very slightly.

The barycenter.

somewhere between the stars