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Diodes have the following properties:

A diode is represented with the following symbol:

The diode above will pass current flowing from left to right, but block current trying to flow from the right to the left-hand side. The voltage drop over diodes will vary from diode to diode, the most common ones are:

"Normal" diodes: Have a voltage drop between 0.6V and 0.7V
Schottky diodes: They have a low voltage drop (about 0.1V to 0.2V)
Light emitting diodes (LED): These diodes will emit a light when passing current. Their voltage drop will depend on the colour of the LED, but is around 2V to 3V.
Zener diodes: They behave as a normal diode for current flowing from the left to right, but will allow current to flow from the right to left if the voltage is high enough. You get a lot of different Zener voltage diodes, e.g. 4.7V and 12V.

Using diodes

The voltage drop across the diode is 0.6V (this is a normal diode and 0.6V is close enough.) The diode and resistor is in series, thus the current through the diode is equal to the current through the resistor.

Thus the current through the diode equals 9.1mA

Here is an example with an LED (you will notice the symbol of an LED has arrows pointing away from it, this is to show that it emits light)

Let's say the LED is a red LED with a voltage drop of 2V, thus:

The more current an LED's conducts, the brighter it will be. To increase the current you can either increase the voltage or decrease the value of the resistor. Just remember that there is a limit to the amount of current a diode can conduct before it breaks and for LED's this is rather low (about 20mA typically)

Zener diodes are mostly used to create a constant voltage. Say we want an LED to be the same brightness, no matter what our input voltage is (the input voltage could be from a battery that starts out at 14V and goes down to 9V before it is charged again)
The circuit below is one way of doing it:

R1 will determine the amount of current flowing through Z1 (for Zener diodes to work correctly, you need a minimum current flowing through them, about 10mA is a good value)
If the input voltage is 9V:

If the input voltage is 14V:

Because the Zener diode will always have a constant voltage drop over it, the potential at Vzener will always be 4.7V. The current through D2 is (assuming the voltage drop over D2 is 2V):

And this current will always be the same, whether the input voltage is 14V, 9V or anything in between. Of course when the input voltage drops below 4.7V, the current through D2 will drop.