All chemical bonding can be explained with electrostatic forces. Opposite charges attract each other. A bond is not a physical object made of matter. It is a force of attraction.
In ionic bonding there is a transfer of electrons. A metal atom transfers its outer electrons to a non-metal atom. The metal atom becomes a positive ion and the non-metal atom becomes a negative ion. The strong electrostatic attraction between the oppositely charged ions is the ionic bond. Each magnesium atom transfers two electrons to an oxygen atom. The oxygen atom gains two electrons, so it becomes a negative ion, and magnesium oxide is held together by ionic bonds.
In covalent bonding there is a sharing of electrons. Two atoms share a pair of electrons. The shared pair is negatively charged, and it is attracted to the positive nuclei of both atoms. This electrostatic attraction holds the atoms together. In a chlorine molecule each atom shares one electron, so each atom ends up with a full outer shell. The nuclei of the atoms do not change in either type of bonding.