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Gravitational force exists for each and every body, whether small or big in terms of size and density.  However, the magnitude of force is very small in case of small bodies.  For masses of size like the planet moon and Earth the attraction force is sizable and can be felt.  Further, we are attracted by many objects on the Earth and so some of these effects cancel out.

The gravity is directly proportionate to the masses of the bodies attracting each other.

for example the force between two persons of 100 kg each, standing 1 meter away from each other will be about 1 millionth of a Newton.  This is negligible to even to move a feather also.  Earth's gravity will take precedence over the interaction of the two people.

But if we take two moons placed some kilometers away, then they will attract each other with a noticeable force.  You can see that meteoroids some time fall on surface of earth.  This is due to the gravity.

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