With the moons gravatational influence on the earth and visa versa, is the center of roaation of the earth somewhere between the earth and the moon or on the earths side opposite of the moon

The centre of gravity of the Earth Moon system is called the Barycentre; this is about 1000 km from the centre of the Earth in the direction of the moon- ie about 6000 km beneath the surface of the Earth directly under the Moon.

This is the point that the Earth -Moon system orbits around, and this point is in orbit around the Sun;
the Earth itself spins on its own axis every 24 hours of course.

I like 'barycentre', but where did you find those numbers?

I cant seem to picture it correctly. If there were just earth. then the revolving moon were presented then the earth would be shifted in the direction of the moon; and as the moon revolved around the eaarth then the earth's center would travel in a small circle correct? So is the ce=nter of this small circle away from the moon i.e. is the earths center between the moon and the circle's center or is the circle center on the other side; and is this the center of the mass of the two moving bodies

This picture shows it very well:

http://baby.indstate.edu/gga/gga_cart/barycen1.gif

Keep in mind that C is the center of mass ... or the Barycenter

The center of mass of the earth-moon system, right?