A semiconductor diode consists of a semiconductor PN junction and has two terminals, an anode (+) and a cathode (-). Current flows from anode to cathode within the diode (according to the high to low circuit analysis method), but only when there is at least a certain amount of forward voltage applied. When positive voltage is applied across the diode, it is called a forward bias, whereas a negative voltage is called a reverse bias.
A diode is best described as a one way valve, since it only allows current to flow from anode to cathode. For example, if you applied a reverse bias to the diode with a magnitude of 5 volts, current would not flow. If you applied 5 volts with a positive bias, current would flow.
This seems pretty simple, but there are exceptions to the one way valve analogy. For example, diodes have a minimum forward voltage level to allow current to flow. In most cases, about .7 volts are needed to trigger current flow. You can see this from figure 1.3 below. The current does not start to flow until a certain amount of forward voltage is applied.
Another exception is the breakdown voltage. All diodes have a point where, if the reverse voltage is high enough, the semiconductor structure will break down, allowing current to flow. This value is usually fifty volts or higher and when the breakdown voltage is reached, it generally damages or destroys the diode.
Why is a diode useful? Because it can be used for rectification, protection of components from reverse voltage, and creating interesting wave shapes. For example, say you have an electrolytic capacitor that can only withstand 10V of reverse bias voltage. All you have to do is place a diode in front of it and it will block most reverse voltages from destroying the capacitor. Rectification is the process of converting an alternating current signal into a direct current signal and is used in all AC to DC converters and power supplies.
An ideal diode's operation, shown above, would cause full current to flow immediately upon application of forward voltage. However, this is not the case in the real world, and with real diodes the forward current increases with additional forward voltage and there is also a limit to reverse voltage before the diode breaks down and allows reverse current. The operation of a real diode is shown below.