You're fishing for "gravity", but you only have half of the story.
Gravity is responsible for the two forces that pull an object and the earth
towards each other.
The gravitational pull acting on an object towards the earth's center.
Gravity pulls objects towards the center of the Earth.
everything, but most importantly the Sun.
Weight is the measure of the earth's pull of an object.
The accepted value of gravitational acceleration at sea level is 9.81m/s2 towards the centre of the earth.
When falling to the ground (or even just in the air), the acceleration of an object depends on the gravitational pull of the object it is falling towards. Here on earth, all things fall relative to the earth which causes an acceleration of 9.81 m/s^2
Force applied towards you (pull) or force applied towards the object (push)
Because the gravitational pull of the Earth is much larger than the gravitational pull of lets say... a human. You could argue that on earth there is a gravitational pull to humans, but the Earth is over powering that pull. So the object is pulled down (to the surface of the Earth) rather than to you or me.
when there is a pull of an object towards the grand
Yes. The Earth and then Sun pull each other towards each other.
A pull towards earth's center
The huge mass and density of the earth have a pull in objects in space. It is still unsure if Earth's magnetic field has any effect on this attraction.