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A Zener Diode (at low voltage) is much like a normal diode. Like a normal diode it conducts at 0.6 volts in one direction and does not conduct in the other direction.
In the direction it does not conduct, as you approach the "zener" voltage it starts to conduct. A 12V Zener conducts at 12V.
Zener diodes are often used to regulate a voltage. It keeps the voltage held down to the correct voltage.
Think of it as a voltage clamp (shunt regulator). As you gradually increase the voltage applied to a Zener (positive to the Cathode, negative to the anode), you reach a level (knee) where a slight increase in voltage causes a large increase in current. That is the Zener voltage that it is rated for. To create a shunt regulator, you also need to have a resistor between the cathode and the positive side of the unregulated voltage source.
Notice that no current flows through the Zener while the input voltage V(unreg) is between ~-0.5 and ~9.5V. For negative input voltages, the zener acts like a normal Silicon Diode. For positive input voltages, the Zener clamps the voltage at ~10V.
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