I understand how a transformer works with the AC electricity and the iron core and the coil ratios. What I don't understand, is what the fifth or sixth wire does. How do you incorporate it in a circuitt, and can you show me a simple circuit of one?

A five-wire transformer *might* be a center-tapped secondary. One coil has a wire in the center, so that you have a choice of normal voltage, or half voltage. Some transformers have taps that are off-center, for other voltages.
http://www.tpub.com/neets/book2/5d.htm

A six-wire transformer *might* be three-phase. Three smaller transformers wired together.
http://www.twincomfg.com/Rectifiers/Twinco_3PhaseWiringSchematic.html

Even a 4-wire transformer may not work the way you expect. Best to have a diagram / specifications from the maker, or test the transformer at (hopefully) safe voltages and currents.

Five wires implies a center-tapped secondary; six wires implies dual secondaries. Both of these were common in vacuum tube circuits.

If there is not a schematic diagram etched on the case or nameplate, it is going to be difficult to figure out. Look for the manufacturer's name and model number, and try contacting the maker.

As a very long shot, the primary winding was usually black and white, and the secondary was red and green. Center taps were yellow, and, in the case of a dual secondary, try blue/yellow for the lower voltage and red/green for the higher voltage.

Another trick is to look for a device with the same transformer or a similar one, and trace out the wires. The lower voltage secondary was for filaments, and the higher voltage for working voltage. Caution. In radio transmitters and TV sets in the vacuum tube era, the high voltage winding were LETHAL voltages.

The absolute last resort is to open the case and try to dope out the windings. It helps to have somebody with you who knows what to look for.