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Two bodies locked in orbit each move about "a common center of gravity" for the pair.

Since the Sun is sooooo much more massive than the Earth, the Sun-Earth common center of gravity lies INSIDE the sphere of the Sun. From a PRACTICAL standpoint, since the Earth's motion about the center of gravity is about a ZILLION times more than the Sun's, it's OK to say that the Earth is circling the Sun.

HOWEVER you CANNOT be so sloppy with this as the two bodies in question start to have similar masses. For example, the Earth-Moon center of gravity is STILL inside the sphere of the Earth, but only just so (about 3/4 from the center to the surface) The Sun-Jupiter center of mass lies OUTSIDE the sphere of the Sun (by about 28 thousand miles or so). So, it is ABSOLUTELY accurate to say that you can think of the Sun as going around and around a circular path due to Jupiter. For binary star systems, the effect is even MORE pronouced, as the center of gravity approaches the center of the physical distance separating the stars.

(Yes, I know I said "circle" when "ellipse" would have been more precise, but we have verbs like "circling" but we don't have verbs like ellipsessing...)