Density is a measure of how much mass is within a 3-dimensional area. For example, water has about 1g per cubic millimeter.
Gravity is a function of mass, where an object attracts other objects based on the amount of stuff inside it.
But gravity is a fairly weak force, and the further away the second object is, the less gravity pulls it towards the first, larger, object.
So, when you take two objects with the same mass, but one is the size of the Earth and the other is the size of the Moon, then standing on this very dense moon will have more gravity than standing on the relatively less dense earth, because you are closer to the center of gravity.
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