Ian G.'s comment is a good starting point for this answer. As
Courtney Seligman, an astronomy professor, observes:
"The Earth and Moon are each moving around the center of mass of the
Earth-Moon system. Since the Earth is much more massive, it is much
closer to the center of mass, which is actually inside the Earth."
Prof. Seligman notes that the Earth and the Moon ""each exert a
force on the other which, according to Newton's Third Law of Motion
(the Law of Action and Reaction), is equal and opposite to the force
that the other is exerting on it." However, "although the forces are
equal, their effects are not, because the more massive Earth is
accelerated less by the same force, than the less massive Moon."
"Gravitational Interactions of the Earth and Moon"
Personal website of Courtney Seligman, Author / Professor of Astronomy
For further explanations of the Earth-Moon system, see:
"Question #1526" (answered by Louis A. Bloomfield)
How Things Work
"The Moon's Motion," by Dr. Jamie Love
In short, the more massive Earth does not move as much around the
barycenter as the less massive Moon.
As your question indicates, an orbit is a result of gravity. Gravity
is a complex phenomenon, still being studied. However, for purposes
of this answer, I think that it suffices to say that a massive object
warps space-time, causing other objects to move towards it (rather
than, say, continue in a straight line).
"Gravity" (27 Dec 1997)
The Why Files
As explained by Dr. Jamie Love in the document cited earlier, the
Moon, in orbiting the Earth, is essentially moving toward the Earth
but never hitting it. (And likewise, the Earth is moving toward the
Moon and never hitting it -- though its movement is not as great,
since, as noted before, it is much more massive.)
For more information about gravity (and related subjects), see:
"Special & General Relativity Questions and Answers," by Dr. Sten Odenwald
Gravity Probe B
Please let me know if you need any clarification of this answer.
Search strategy --
Searched on Google for:
"earth orbit the moon"
"earth orbits the moon"
earth moon barycentre
earth moon barycenter