The basic underlying nature is quantum mechanical, matter is composed out of electrons and nuclei, nuclei are composed out of protons and neutrons,( protons and neutrons are composed out of quarks and gluons.) That is as far as we have come in analyzing matter in the microcosm.
I can't understand why the centers of the two types of charges are slightly separated.
In the hydrogen atom, the electron is in an orbital in a quantized energy state around the proton, usually the ground state. Orbitals are the mathematical description of what you call "electron cloud". They describe the probability locus of where the electron can be found if it is looked for. In the link illustration it is placing extra electrons , creating negative ions of hydrogen.
In solids there are very many electrons in the atomic structure and also atoms are usually bound up as molecules. The probability distribution of the electron locations and their accompanying charge is usually structured due to the quantum numbers of the orbitals, as seen here:
Suitably aligned f atomic orbitals overlap to form phi molecular orbital (a phi bond)
Where the electrons are, the positive fields from the nuclei in the center are masked, but in neutral solid state lattices, these structures are symmetric, and by necessity create volumes where the negative charge distribution of the electrons allow the positive electric fields of the nuclei of the molecule to pass unimpeded.
The negative charge density follows the Ψ*Ψ probability distribution, the orbitals.
In a neutral atom/molecule the center of negative charge will be calculated by averaging the orbital shape, for the electrons, similar to finding the center of mass of a massive body.
The positive charge density depending on the number of atoms, will have a geometric center of charge at the lattice point ( the nucleus is like a dot, see the hydrogen link) . Since the body is neutral and symmetric the negative charge center of the electrons probability cloud and the positive charge center of the nuclei should overlap.
So there are positive and negative volumes symmetric around the center of positive charge of the nuclei of the molecule, and lego like, bonds happen, which create solids by attraction of positive regions with negative regions, generating a neutral lattice of a solid.
When the rod field introduces an asymmetry, the positive and negative charge distribution will have a different location and will be separated, the negative center of charge will be pulled by the positive rod ( or pushed by a negative charged one) leaving the positive(negative) electric fields on the other side.This is a distortion of orbitals, leaving the other side open to the more positive charges of the nucleus,molecule by molecule.