Zener diodes allow current to pass in the normal direction just like any normal diode and with the 0.6V ish drop, again just like any normal diode. Whats different about them is that, unlike a normal diode which blocks current in the reverse direction (until you hit its reverse breakdown voltage), a zener diode will block current in the reverse direction until its specified voltage is reached and then it will allow current to flow. This means when used in reverse the zener will cause a voltage drop of a set amount (roughly – the zener voltage will vary depending on conditions, tollerance etc) in a similar way to the 0.6Vish dropped by a normal diode operating in the normal forward flowing current mode.
Vz = The zener votlage (the voltage drop across the diode)
Iz (or Itest etc) = The current passing through it. This is the current that gives the other specifications
Zz = Impedance / AC Resistance. A zener has an internal resistance, think of is as a series resistor inside the diode. Its a dynamic resistance – it changes with current, temperature etc.
Power – Remember your zener will heat up by the amount of power it is dissipating, just like say a linear voltage regulator!
Calculating your resistor value to place in series with the zener to give you the zener voltage out:
Resistance = (Vin – Vz) / Iz
So for a 1N4733 say, (12V – 5.31V) / 50mA = 138ohms
You could then remove the Zz from the resistance you use, but often you don't bother as its typically low and won't affect things much.
What happens when you change the current
Additional current – Increase input voltage but with same external series resistor = increase in current through the zener diode = more voltage dropped across its internal resistance = increase in output voltage. This is one reason why they often aren't great replacement for a linear voltage regulator, because their output voltage will change with input voltage. Other reasons are that you have to throw away current through them to get their output voltage plus the circuit changes dependent on the load if its variable. That said, if your input voltage is fixed and your load current is fixed then a zener can be a great choice in circuits as you can then select your Vin series resistor based on the load current and zener current you want and know it will not change.
Reduced current – zeners don't work down at uA (Vz falls), they need a bit of current to do their job. However at lower currents of mA they will typically work ok.
Using zeners for clamping, protecting inputs, etc
Zeners can be great for this as long as you don't mind needing to add a series resistor before them – so good for signals not good for power inputs. You just need to size your series resistor based on what you expect worst case inputs voltages to be be (to ensure the zener power rating isn't passed) and for lower voltages to ensure enough current will flow through it to do its job properly. One other advantage of zeners in this applicatton is that because a zener acts like a normal diode in the other direction it will clamp negative voltages at -0.6V naturally.