As you learned on the mechanics of the PN junction, when a negative voltage from anode to cathode is connected, the electrons do not have enough energy to cross the widened depletion zone and no current flows.
But what happens when enough voltage is applied, the electrons have enough energy to cross the barrier and knock some other electrons free along the way. This creates what is called an electron avalanche where more and more electrons break free and conduct current. The voltage at which this phenomenon starts remains across the diode constant even if the outside source is increased.
In most semiconductor diodes this effect is destructive to the diode, since they are not designed to handle the current produced by the avalanche.
The zener diode works in what is called the zener region, a voltage where a small and controlled electron avalanche can be used to generate a constant voltage across the diode.
This property gives the zener diode many of its uses as a voltage regulator, voltage monitor and many other fixed voltage applications.