Here's everything you'll need:

A toy washing machine! (...Of course)
In this guide, we modify a:But we've also modified a:
Miele SoftronicCasdon 476 Hotpoint Electronic Washer

A nice chunky motor
We used this drill motor, which is quite small but nice and powerful!
(Click it for a full-size view)
It is rated at 6 to 15 Volts DC. Both of the example toy washing machines above use a power source of 4.5 Volts (3 C batteries), which is nowhere near enough.
So, we're also going to need:

A suitable power source for the new motor
I used a regulated multi-voltage power supply set to 15 Volts (its highest setting), which can supply up to 4 Amps (I know on the YouTube video I said 5, but I secretly meant 4... >_<). As well as voltage being important, you also need a power source which can supply a decent amperage. This determines the power of the motor. At high voltage, the drum might be able to get to fast speeds, but only when it doesn't have a heavy load in it! I really wouldn't suggest anything below 300 milliamps.

An n-channel MOSFET
This is like a big transistor. This is what gets the pulses of electricity from the computer. When it gets power, it switches off a circuit (that will be our washing machine motor and power source!), and when it loses power, the circuit is switched on again. You can think of it as a sort of 'translator' between the pulses which are too low or too high voltage for the motor, into the correct voltage.
Don't worry if you have no idea how to wire one, I'll explain it later. ;)
MOSFETs are not expensive (costing no more than about 2 GBP, 4 USD, or 3 euros).
Most MOSFETs can handle switching a large voltage (over double mains voltage!), but before you buy one, make sure that it's capable of switching the voltage you will be using with your new motor. If that voltage is less than 12 Volts, make sure the MOSFET can handle 12 Volts, because that is the voltage which the computer gives out.
IMPORTANT: Make sure it is an 'n-channel' type! A different type would have its 3 terminals assigned to different things, meaning that we'd wire it up not how it should be, so it wouldn't work, and would cause damage to the MOSFET and possibly to the computer.

Some wire
We won't be needing more than about 1.5 meters of the stuff, but it would be better to get 3 meters or so to be on the safe side (it's not like it costs much). As you can see in the picture of the motor (above), the wires don't need to be thick (but make sure they're not too ridiculously thin, or you run the risk of the wires melting or burning out).

A serial cable (Female to Female)
Since we're going to be using the computer to control it, we need a cable to join between the computer and the washing machine. These cables are often referred to as 'RS-232' or 'DB9' serial cables. Make sure it's long enough so that the washing machine can be above a bowl or something to drain the water into.
(Click image for full-size view)

A serial plug (Male)
(Don't worry, we're only going to use 2 pins on it!)
From my testing, I found these hard to find on the web. But you can get this from Maplin, where it's called a 'D-Range 9-Way Plug'. Make sure you get the plug rather than the socket, because the socket is female. Either that, or you can get a Female to Male serial cable and a 'D-Range 9-Way Socket' (which is female).

Once you've got all the stuff, let's open up the washing machine! --> Section 3 - Opening washing machine