Dear Student
The forces that the Earth and Moon exert on each other must be equal, but that does not mean that the effects of those equal forces are the same, because the two objects have very different masses.
Earth exerts an gravitational effect on the moon that is 80 times stronger than the moon's pull on Earth.
Hence, The moon orbit around the earth due to gravitational force of the earth. This is the same effect which causes Earth to orbit around Sun.
The Moon's gravity pulls on the Earth and makes the tides rise and fall every day.
Regards