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Every solid object, whatever its shape, has exactly one center of gravity -- a point at which the entire mass of the object can be concentrated insofar as gravity is concerned.  Suspend the object at its center of gravity and it will be balanced there in any position without a tendency to rotate.  Toss the object aloft with any amount of spin and it will always rotate exactly around its center of gravity. 

We all understand these things, and... 

Oops: a tossed object, while falling, does not experience gravity -- rather has its weight balanced by acceleration, making it "weightless."  Thus, the spinning of an object around its center of gravity has nothing to do with gravity.  Same for objects in space.  Indeed, the phrase "center of gravity," which is so familiar to us all, may be something of a misnomer.  For this puzzle, we shall prefer the phrase "center of mass," which, like ‘mass’ itself, maintains its full meaning for an object anywhere in the universe, with or without gravity.  Celestial bodies -- stars and planets and moons -- each spins unendingly and unerringly around its own center of mass. 

tars and planets and moons do something else, too.  They do a thing called orbiting.  Our little old solar system has thousands of bodies orbiting around the sun.  Most of them are asteroids plus some small number of planets and a some comets.  Meanwhile moons and rings do their orbiting for the most part around the planets.  Often pairs of celestial bodies are seen to orbit each other.  Of course, when one is much more massive than other, the "other" appears to do all the orbiting.  In reality, though, both bodies are orbiting around their common center of mass as if they were one object.  That point is called their "barycenter." 

For example, the moon appears to orbit the earth every 29 days at a distance of 239,000 miles.  Our planet, which has a circumference of about 25,000 miles, is 81 times as massive as the moon.  Suppose you wanted to depict them both in, say, a magazine fold-out and show the location of their barycenter.  Your illustration might look something like this...