With Zener diodes, it's better to say that when you apply a reverse bias voltage that's higher than the Zener voltage, then a current will begin to flow, and once there's a current flowing the votage drop across the Zener won't change if the current changes. The P-N junction is heavily doped (lots of holes and free electrons) and so when the potential rises above the Zener voltage electrons from the P side will start doing their quantum-mechanical-magic-that-I've-never-fully-understood and tunneling through to their friends on the N side, the opposite of how they're "supposed" to behave. The potential energy (voltage) that the electrons need to tunnel is essentially the same regardless of how many electrons are tunneling, so the voltage drop can stay constant as the current varies.
Zener diodes always have to be used with some kind of current limiting, they're kind of like an LED in that regard, since the device current increases extremely rapidly with increasing voltage. If you just stick 10 volts across an LED or Zener that has a forward voltage of 1.2 volts or 5.6 volts respectively it's going to go "no thanks" and burn out. All Zeners have a maximum current rating and therefore a maximum power dissipation, and a minimum current where the regulation gets bad. If you had a 5.6 volt Zener connected to a 9V power supply through a 220 ohm resistor, the Zener would pass about 15 mA. If the power supply voltage increased, there would be an increase in current through the resistor and Zener, but the voltage across the Zener would stay the same. The "excess" voltage would both cause an increase in current and more power to be dissapated as heat in both the Zener and series resistor.