When it is said that charge "moves" throughout an object, like if negative charge moves to the edge of an object and the charges become polarized, does this mean that the electrons has moved to the edge of the object, or does it mean that negatively charged atoms moved to the edge of the object? Or does it mean that the actual magnitude of the charge contained in electrons moved to the edge of the object? If you have a negatively charged balloon and move it next to a piece of metal, the electrons are said to repel the electrons in the balloon and move as far away as possible. Does this mean that the electrons literally detach from their atoms and move away, or does it mean that the atoms containing the electrons moves away as well?
Taking copper (atomic number 29) as an example of a good electrical conductor in the solid (and liquid) state.
On average there are positive copper ions Cu$^+$ with 28 orbiting (bound) electrons and for each ion one free/mobile (unbound) electron.
When no external electric field is applied these free electrons having thermal energy move about at random throughout the metal just like the molecules in a gas.
In the solid the positive copper ions are fixed in a lattice and vibrate about a mean position.
When an electric field is applied the free electrons can move under the influence of the electric field. It is these free electrons which result in copper being a good conductor of electricity (and heat). The positive ions are fixed into the lattice and so can only move their mean position very little.
If the final state is such that the net movement of the free electrons is zero (electrostatics) then the redistribution of charges throughout the metal is such that the electric field produces by the induced charges as a result of the movement of the free electrons is equal and opposite to the externally applied electric field and so the net field in the conductor is zero.
So when you see diagrams of conductor with positive and negative charges on them, the region labelled as having a negative charge is one where there is a net surplus of free electrons and the region and the region with positive charges is one which has a net deficit of free elctrons.
Usually the electrons are the more mobile charge carriers, so in most instances where "charge moves", it is the result of motion of electrons. But in a particle accelerator where you accelerate protons, the charge that moves is a proton, not an electron...
Depending on the structure of an object, if it's a solid that means almost by definition that the atoms have a fixed location in the object, so that polarization is caused by the motion of electrons. If an object ends up positively charged, it is because electrons were removed; if it's negatively charged, it's because electrons were added. It is much easier to add an electron than to remove a nucleus...