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First, the measurements of volume, weight, and mass are different. This is important.
Volume is the amount of physical space something takes up. It's measured in cubic metres (as well as litres, gallons, etc. for liquids), like the units you have given.
Mass is the inherent property of matter, while weight is the force applied to a mass by the Earth's (or some other large body, like the Moon's) gravity. On Earth, weighing scales measure in Newtons, weight, but do conversion to give you conventional grams and kilograms, units of mass. Mass is the property of matter, while weight is the force produced by gravity.
Density is, in a nutshell, mass per unit of volume. Water is around 1000 kg / m^3, meaning there is 1000 kilograms of mass for every cubic metre of water. Different materials have different densities, and this is because of all sorts of things. The atoms might be very tightly bonded together in a lattice, like in many metals. This is why metals are usually very dense. While they could also be free-floating molecules with very weak attractions. This is why gases are not very dense.
In the case of gold vs copper, the former has much larger nuclei with much more protons, and that probably contributes to an increased amount of attraction in the metallic lattice, making gold denser than copper. This means the gold will be heavier than the same volume of copper.
It's not just about the number of protons though! The configuration of electrons is also very important, arguably more so than proton number. For instance, osmium and iridium are both denser than gold, but they have smaller nuclei. Electron shell shenanigans are everywhere in chemistry!