Oxidation is the loss of electrons. Reduction is the gain of electrons. A memory aid is OIL RIG: Oxidation Is Loss, Reduction Is Gain.
Look at the half equations. Mg → Mg2+ + 2e- shows magnesium atoms losing electrons, so magnesium is oxidised. Cu2+ + 2e- → Cu shows copper ions gaining electrons, so the copper ions are reduced. In Na+ + e- → Na the sodium ion gains an electron, which is reduction.
Oxidation and reduction always happen together, because electrons lost by one particle must be gained by another. This is a redox reaction. In Zn + Cu2+ → Zn2+ + Cu, zinc loses electrons and is oxidised, while the copper ions gain electrons and are reduced.