center of mass
Balance a ruler on one finger and there is a single point where it sits level — that is its center of mass, the average position of all its weight. For a whole orbiting system the same idea holds: two bodies bound by gravity each circle a shared balance point that hangs in space between them. In astronomy this point has its own name, the barycenter.
The center of mass sits closer to the heavier body, in the same proportion as a seesaw: if one mass is ten times the other, the balance point is ten times nearer to the heavy one. Crucially, with no outside force the center of mass drifts in a perfectly straight line at constant speed no matter how wildly the two bodies whirl around it. That is why splitting any orbit into 'motion of the center of mass' plus 'motion about it' is the master move that makes the two-body problem solvable.
Barycenters are not a bookkeeping trick — they are observable. The Sun itself is tugged into a small wobble about the Solar System's barycenter by Jupiter, and a star wobbling about its barycenter with an unseen companion is exactly how the first exoplanets were found. The Earth and Moon orbit a barycenter that actually lies inside the Earth, about 4,700 km from its center, so the Earth genuinely wobbles each month, it just does not swing clear of itself.
Pluto and its big moon Charon orbit a barycenter that floats in empty space above Pluto's surface, so the pair waltz around each other like a dumbbell — the clearest planetary example of two worlds truly circling a point between them.
When neither body dominates, both visibly circle the barycenter.
Center of mass and 'center of gravity' coincide in uniform gravity but are subtly different concepts; in astrophysics the mass-weighted center is what matters, and it is fixed by the masses, not by which body we happen to call the 'central' one.