Like positive and negative charges attracts each other , any two atoms can attract each other even when they are electrically neutral.But the force with which they attract each other was extremely small when compared to force between two charges.(in the order of one millionth of the force of attraction between charges) . so you might be wondering that why there was no force between two objects close to each other.The answer to this question was the force between two objects is proportional to product of their mass.As I said before,this force is extremely small,you can't see its effect unless the mass involved is very large. since the mass of earth is relatively large , the force between earth and any other object can be perceive.So any object in the vicinity of the earth will be attracted towards the center of the earth.At the same time, the object tries to pull the earth towards it with the same force the earth pulls the object. so why the earth wins i.e why object was moving towards earth and the earth was not moving towards object ? The answer to this question was since the mass of the earth was very large,it will take more time to accelerate towards the object. Than the time taken by object to move towards earth.so you see the object falling to earth...
