How gravity behaves for an Earth body and for a celestial body? Is there any difference in the gravity of the Earth for an Earth body or a celestial body? For example, will gravity be same for an Earth body, compared to the Moon?
Gravity is the same qualitatively.  It is the same type of thing: It pulls you down.  It can be described by Einstein's equations.  It can be very closely approximated my Newton's equations.  Force of gravity is proportional to mass and and inversely proportional to distance²

Newton proposed a law of Universal gravity.  The same law of gravity applies everywhere.  Although the details of what the law actually is have been tweaked (by Einstein) the principle of "the same law everywhere and at all times" seems to be true.

Gravity is different quantitatively. It is stronger in some places and weaker in others. The moon has a much smaller mass, and so your weight on the moon (ie the force of gravity on you) would be much less.