You can use plumb lines to find its center of gravity.
Hold it from one point and draw vertical line in direction of gravity (plumb line). Hold it from another point and do the same, and a third point if necessary; where the lines cross is center of gravity
Secret
Mass divided by Volume = Density. Or use the displacement method for an irregularly shaped object.
It is the displacement method.
Google "Archimedes' Principle" and see if that gives you a clue...
Half fill a graduated glass beaker with water and note the level. Place irregularly shaped object into the beaker. Note the new level. The difference between the two levels is the volume of the object. cw: OK, that method works for objects that don't get wet -- like steel. Tissue paper would not be a good candidate. Also, if the object floats, you have to submerge the object.
Finding the center of gravity of an irregularly shaped object can be done using the plumb line method.
ways of locating center of gravity?
Secret
Friend Hilmar Zonneveld is perfectly and absolutely right. Weight, being a vector, of an object will always act through the center of gravity. Also definition of centre of gravity confirms that whatever be the position the weight would always act through a point known to be center of gravity
the water displacement method
What you are asking is quite clear, but the wording is a little misleading. Every object containing any amount of mass, however great or small, has gravity. So the answer to your question will essentially be the answer for any massive body whether it is a planet or a bowling ball. Gravity is a property of mass. Since every little bit of the planet earth is exerting its gravitational effect on you, then there must be a place within the earth that is the point where all the individual effects 'average out'. That point is the 'center of gravity'. If you mentally take any solid object that is irregularly shaped (but not too strangely shaped for a first go at this) then you can do the following. Suspend the object from a point on the surface and notice how the object settles. Imagine a perfectly straight line going down through the object from the point of suspension to the earth's center of gravity. Now choose any other suspension point for the object and do it again. The two 'lines' formed in the object will meet at one point. No matter how many times you suspend the object freely from a point on its surface, the lines going down through the object will all meet at one point-- the object's center of gravity. [With very irregularly shaped objects, this point might even be outside of the object itself!] If you could somehow suspend the object perfectly on a tiny ball bearing right at that point, the object would move and turn freely, without any part of the object tending to settle toward earth. The center of gravity would be a perfect balance point for the object. Because planets and moons are naturally formed, their materials are not perfectly smoothely distributed within them. So centers of gravity are usually not the very same point as the 'geometric center'. The same is true for a black hole. It may even be simpler for black holes. The singularity, the center of gravity and the 'geometric center', whatever that means for black holes, would probably always be the same point.
Mass divided by Volume = Density. Or use the displacement method for an irregularly shaped object.
Asteroid
If the object in question is water proof, and small enough. You can measure the amount of displaced water when it is placed in a tank full of water, this may not work for a lot of things but it can help when attempting to measure some objects.
Tight
Use volume displacement method.
Take an amount of water of which you know the volume. Drop the object in the water. Find the difference