What is "forward" and "reverse" voltage when working with diodes?

The forward voltage is the voltage drop across the diode if the voltage at the anode is more positive than the voltage at the cathode (if you connect + to the anode).

You will be using this value to calculate the power dissipation of the diode and the voltage after the diode.

The reverse voltage is the voltage drop across the diode if the voltage at the cathode is more positive than the voltage at the anode (if you connect + to the cathode).

This is usually much higher than the forward voltage. As with forward voltage, a current will flow if the connected voltage exceeds this value. This is called a "breakdown". Common diodes are usually destroyed but with Z and Zener diodes this effect is used deliberately.

Forward-bias is when the anode (the pointy part of the symbol) is positive and the cathode (the bar) is negative. Reverse-bias is when the anode is negative and the cathode is positive. A lot of current flows when the diode is forward-biased, provided that the voltage is higher than 0.6V or so for a silicon diode or 0.3V or so for a germanium device. A very small amount of current flows if a diode is reverse-biased.

If you have a DVM and some diodes, you can check it for yourself. Diode cathode leads are usually identified with a band, so if you switch the DVM to a low resistance setting, and connect the leads across the diode in both directions, you should see a low resistance in one direction and a high resistance in the other direction, provided that the DVM is supplying a high enough voltage. Some DVMs have a special diode test setting that is easier to use.

LEDs usually have a flat against the cathode lead.

Typically the forward voltage is the voltage at which current starts to flow in the normal conducting direction (as mentioned above it's somewhere in the range 0.3-0.6v)

Reverse voltage is sort of the same thing - it's the voltage where current starts to flow when the diode is in the normally non-conducting region - this is also the point where the diode is likely to turn into a charred mess as all the internal semiconductor stuff turns to mush (choose a value somewhat larger than the largest PEAK [not RMS] AC voltage the diode will see)