Question: I am new to physics and I understand *gravity* is a pulling force.

Let's assume a *ball* has been placed on the ground. Does *gravity* has any role after I placed an object (e.g. ball) on ground? 

Sorry, if I asked something silly or the question was wrong.

Answer: 
The ball is in an equillibrium of two forces: the gravity and the pressing force from the ground. 

You can say, that the two cancel out. But there is, in practice a difference, if you have no forces acting or some forces that cancel out. Because usually the forces change, if you change the system. In your example, the force from the ground gets smaller if you lift the ball a litte, because it is sqeezed less. The ball takes such a position, that the two forces are equal, because it experiences a net force and gets moved as long as they are not. 

I hope the difference between cancelling forces and no forces is clear? - the former makes possible a stable system. With no force the smallest excitation would set the ball moving (with time arbitrarily far).
The force of gravity never stops acting on the object. However, the gravity is balanced out by an equal and opposite normal force. Gravity simply keeps the object stuck to the floor. It does not stop existing; it is merely balanced and therefore has no overall impact on the object's motion. Hope this helps! -Alm.