The Earth around the Moon We know that the moon rotates around the earth  due to gravitation. But both moon and earth attract themselves towards them. So why doesn't earth goes round the moon? It is feeling a force too. So it should also accelerate around the moon.
 A: It does. The Earth and Moon both rotate "around each other" as modelled here (from Wikipedia):

Note that this is just and example model. The effect on Earth is in reality even smaller than visualized here (and the Moon's orbit is not perfectly circular). 
Had the two objects had the same masses, then the orbital motions would have been equal:
 
The issue is what "rotating around something" exactly means. You could say that neither of the objects rotate around each other in any of the cases shown - because in fact they rotate around their shared "mid-point", so to say. This point is called the barycenter. The barycenter is the point that their gravitational influences "average down to", if we were to imagine a stationary non-orbiting object that they both rotated around.
A smaller (less massive) object gives a weaker gravitational pull, thus causing a smaller centripetal acceleration of the more massive object, giving it a smaller orbit and smaller orbital speed. This is the case for the Earth-Moon system.
Although the mechanism is the same, and they both still rotate around the barycenter, the more massive Earth is rather "wobbling" than rotating/orbiting. It is still orbiting about the barycenter, but that barycenter is located inside it not far from it's own centre.
Some good illustrations of the real barycenter location in the Earth-Moon system are found in this answer on the Astronomy SE site, visualizing how close that barycentre is to Earth's own centre and therefore how little an effect the Moon has on Earth.
A: You are right, even in a simple two body system, the two bodies are orbiting around a point between their centres called the barycentre.  In the case when one body is much larger than the other, this point may be inside the larger body.  In this case, it is common to consider that the smaller body is orbiting the larger.  This is the case with the Earth and the Moon but not for Pluto and its moon Charon.  It is the case for the Sun and all of the planets except for Jupiter.  Even a man made satellite and the Earth will have a barycentre that is not quite at the centre of the Earth but only a trivial distance from it.  At the other extreme, you could have a binary star in which the two stars had the same mass, the barycentre would be the mid point between their centres.  
