The earth and the moon are attracted to each other by gravitational force. Does the earth attract the moon with a force that is greater or smaller or the same as the force with which the moon attracts the earth?

Yes the moon do attracts the earth but the earth also attracts the moon. The gravitational pull of earth is too much as mass is directly proportional to the gravity and mass of earth is extremely greater than the mass of moon. So the attraction exerted by the moon on the earth becomes negligible.