The normal diode and zener diode are very different from each other because of doping. The zerner diodes are heavily doped. This creates a narrow depletion region.

When reverse biased voltage applied to the Zener diode is increased, the narrow depletion region generates strong electric field.

When reverse biased voltage applied to the Zener diode reaches close to zener voltage, the electric field in the depletion region is strong enough to pull electrons from their valence band. The valence electrons which gains sufficient energy from the strong electric field of depletion region will breaks bonding with the parent atom resulting in increase in electric current for a small increase in voltage.