Oxidation and reduction can also be described using electrons. Oxidation is the loss of electrons and reduction is the gain of electrons. A helpful reminder is OIL RIG: oxidation is loss, reduction is gain. This idea works for reactions that do not involve oxygen at all.
When a metal atom forms a positive ion it loses electrons, so the metal is oxidised. For zinc, the half equation is Zn → Zn2+ + 2e-. When a metal ion turns into a metal atom it gains electrons, so the ion is reduced. For copper, the half equation is Cu2+ + 2e- → Cu.
In a displacement reaction both changes happen together. If zinc is added to copper sulfate solution, zinc atoms lose electrons and copper ions gain them. The sulfate ions do not change, so they are left out of the ionic equation: Zn + Cu2+ → Zn2+ + Cu. The zinc is oxidised and the copper ions are reduced.
To identify what is oxidised and what is reduced, look for the species that loses electrons (oxidised) and the species that gains electrons (reduced). In a half equation the electrons are written on the left for reduction and on the right for oxidation.