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For small bodies, this and the CG are coincident. For superlarge objects such as the moon, the CG and the center of masses will be close but not coincident.
If the bodies are a sphere and are of uniform density (which the moon is not very close) then the CG and the centre of mass are the same, regardless of whether they are small or superlarge.
1106, normally the stab is set for the C of G. If that is not possible (eg, an aircraft that has no trimable stab), then the elevator trim tab is set to the correct setting.
I hear that the setting is computed for a V2 climb OEI with zero elevator force, though in the sim I've found it not to be all that close.