Diodes are semiconductor devices, meaning they are conductive under certain circumstances and not under others. When a diode has voltage applied one way, it will block current from flowing. Reversing the polarity will cause it to conduct just like a wire. The two ends of the diode are named the anode and the cathode. How is this useful? The first thing which comes to mind is rectification - turning an alternating current into a direct current.
When the voltage on the diode's anode (left terminal) is positive, and the voltage on the cathode (right terminal) is negative, current will flow, making the top of the resistor positive. When the voltage is reversed, with the negative voltage on the anode, the diode turns off, and current no longer flows. This in effect turns an alternating current into direct current. The voltage is not really usable as it has too much ripple, a term referring to the shape of the waveform. To eliminate the ripple, a capacitor may be added across the output to act as storage for the charge.
Diodes are rated by their forward current, which is how much current the diode can have passing through it, and by their reverse breakdown voltage, which is the amount of voltage you can hook up backwards without the diode failing. If you apply a 12 VAC alternating current to the circuit, half of the time the diode has the negative half of the alternating current on it. If the diode can't handle at least 12 V in reverse, it will fail.
Another type of diode is a zener diode. They act like a normal diode when forward biased, but in reverse they breakdown at a predictable voltage and are not destroyed, since they are manufactured to operate in the breakdown region. They are wired backwards, with the positive voltage on the cathode, and must always have a current limiting resistor, since they are a short circuit at any voltage above their breakdown voltage.
Because of this behavior, zener diodes are great for low power voltage regulators, where the current requirement is only a few milliamps or less. Such regulators are not good for higher powers because the resistor and zener diode have to turn all of the excess power into heat. Also, the zener voltage varies based on the current through the diode, so at higher currents the voltage will be higher.
Light Emitting Diodes, or LEDs, are special diodes which emit light as current passes through them. The light is narrow spectrum, meaning it is all about the same color, but it is incoherent, meaning the light is not all in lock-step. This is what separates LEDs from laser diodes.
LEDs have various forward voltages, which is important in designing circuits because the size of the current limiting resistor is based on the current and the voltage. The forward voltage nearly tracks light frequency, with red LEDs having around 1.2V forward voltage and deep blue or ultraviolet having nearly 4V forward voltage.
There is a tutorial dedicated just to LEDs because there is much to know about LEDs, especially if you are going to do something more complicated than just turning one on.