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This doesn't apply to you, you're the Recycler not a mere user! (But mind the Big Capacitor)

There's so much computer stuff being thrown away now that it's worth thinking hard about ways in which it could be re-used. Obviously the best way to recycle things is to actually re-use them as they are and not break them down. There's something to be said for having an old basic machine for word processing which doesn't go on line and is therefore totally immune to advanced viruses or being hacked into. Working systems can also be sent off to underdeveloped countries by charity organisations.
However stuff like 286 or 386 machines, faxes or anything faulty are only good for parts. Stepper motors can be used as tiny electricity generators - the idea's catching on and there's already a page about doing it.

Anyway, let's have a look at the tablefull of stuff that's left over after you've extracted the nice stepper motors.
When you dismantle a piece of old IT equipment, you should break it down as far as possible into three categories for further recycling: plastics, circuit boards and metals. Not a lot can be done with plastics in most areas of the UK at the moment, though if you look carefully on the parts you'll notice they've got recycling codes so the plastic type can be identified. If there is proper plastic recycling where you live you can use the codes to make sure they go to the right places. Circuit boards can be melted down in a high temperature furnace and the precious metals extracted, and the actual metal frames and chassis parts can go into the metals skip. Your local recycling facility will be happy to receive stuff that's already been broken down as it saves them work and improves the quality of their refuse.

You'll need some basic electronics tools; screwdrivers, pliers, cutters, a small adjustable spanner would be handy, and a soldering iron plus a solder sucker for desoldering. A basic digital voltmeter or DVM is an essential so you can test the components. An electric screwdriver is handy if you're in a hurry.

Most things will have a processor board or digital section. Unfortunately there's usually not much you can do with this as the components are specialised and no use in other circuits. You might be able to get a couple of LED's or switches off it, then add it to the pile for recycling.

The power supply's a better place to start looking for useful stuff.
In computer supplies there's usually a cooling fan which is often a 12 Volt one. It will need cleaning with a vacuum cleaner followed by a damp tissue to get the dust off.
To make sense of the power supply, you need to know a bit about how it works. You may need to print this diagram or at least open it in a separate window. Virtually all power supplies will be ‘switched mode’ types. That is, instead of putting the 230V 50Hz (at least in Europe) Mains through a great big transformer and then rectifying and smoothing the low voltage outputs, the actual Mains is put through a bridge rectifier. The resulting high voltage DC is smoothed by a large capacitor, and then a high voltage power transistor chops it up into fast pulses which go through a small transformer with less windings and core metal due to the high frequency, saving size, weight and energy. On the output side of this transformer are diodes which rectify the high frequency AC back to DC again and smoothing capacitors which store it in between the power pulses. Again, as the frequency is so high the capacitors can also be smaller than they used to be.
Once you've looked at a couple of power supplies, you should be able to spot the main bits easily. Here's the board from a small laser printer with the power section in the lower left corner. This is a computer power supply, and here's an older one which had the Mains section on a separate board. Power supplies designed to accept 110V or 230V input have two main high voltage capacitors like this one from an old 386. The voltage select switch connects them in either series or parallel.

Before you start chopping up one of these things, a word of warning. The big capacitor when in use stores a lethal amount of energy at over 300 Volts - that's why you find all sorts of warnings on the power supply case. Built in to the circuit is a ‘leak’ resistance which slowly discharges the capacitor so it should be safe to handle a few minutes after switching off. However, you've usually no idea of the history of the equipment and it's possible a fault could have prevented this happening - the resistor itself could have burnt out or a crack in the circuit board could have disconnected it. The capacitor or capacitors should be easy to spot as they'll be physically the biggest on the board and with the highest voltage rating - something like 150uF at 400 Volts. Before doing anything to the circuit board short out the capacitor's terminals with an insulated screwdriver or pliers. Don't touch the board with your other hand to avoid a shock and keep your face at a safe distance in case of sparks. You should find that there is a disappointing total lack of any sort of spark in which case the board is safe to handle. (If you do manage to get a spark, repeat the process until you don't get one, and ideally check with your voltmeter afterwards to confirm.) Once you've sorted that out it's safe to work on the board. The filter will contain some smaller capacitors with a high working voltage, but they're not big enough to be dangerous - usually less than 1uF. If you're a concerned parent, take the big capacitor off before letting the kids loose on the board - it's the only bit that could be dangerous and not a lot of use anyway.

Having found the capacitor and made sure it's safe, you should find a bridge rectifier right next to it. It's worth unsoldering that. Although it was originally used for Mains voltage, it can be used just as well at low voltage at up to a couple of Amps. Unsoldering takes a bit of practice so if you haven't done much before, have a go at some of the less useful bits before going for the bridge rectifier with its four wires. The great thing about recycling stuff like this is there's so much of it, it doesn't matter if you fry a few things in the process of learning!
Rectifier diodes are probably the best things to start on as they're capable of standing a lot of heat. The most sensitive things are LED's which have softer encapsulation. You don't have to worry about static straps, the power supply components are tough enough not to need them.

The next most interesting area is on the output side of the transformer. The diodes (often marked "d" or "cr") which rectify the low voltage are a special type. To save energy, a lot of them are Schottky Diodes which have a much lower volt drop than a normal diode. Any diode lets current flow one way but not the other, but normal silicon diodes actually need one side to be at least 0.6 of a Volt higher than the other before letting any current through. This doesn't matter a lot at Mains voltage, but a 5 Volt output would need 5.6 Volts in, losing more than 10% of the power straight away! If you get your multimeter on the Diode Test range you should find the Schottky ones only read 200 or so compared to 600 or 700 for a normal diode like this. (The reading is milliVolts. You may also find some that read 400 or so - they're still better than ordinary silicon ones) They're worth getting, as they can be used as pass diodes for a small solar panel or for rectifying the output of a stepper motor when you need a low volt drop. Sometimes they'll be just a normal looking cylindrical diode. Bigger ones are in the form of a flat tab on a heat sink; these sometimes come as a pair of diodes in one case with the cathodes connected together. They can be handy - for rectifying the output of a stepper generator you often need four diodes with their cathodes all connected together. Depending on what you use them for you might not need to bolt them to a heat sink. A Schottky Diode with 0.2V drop at one Amp is only going to get 0.2W of heat which it could stand on its own.
Any ordinary silicon diodes you find are also worth having - use them for applications where you don't need such a high efficiency such as for reverse polarity protection of a solar panel.

Having got the diodes, the next thing to look at is the small smoothing capacitors next to them. They're often rated something like 2200uF at 25 or 35 Volts. If you're making a LED wind-up torch, these capacitors are ideal to store the energy to make it go on after you've stopped winding; anything of 1000uF or more is worth having. Just make sure that the voltage rating is slightly more than the biggest voltage you're going to apply. (The big Mains capacitor is no good for this  - it's the amount of uF that counts, not the amount of Volts it can stand).

Considering these boards are not meant to be serviced and mostly assembled by machine, the designers are surprisingly helpful with printing component identification on them. This one's virtually got the circuit diagram and the connections for all the components on the underside!
Many of the components are also helpfully marked like these single and double diodes.
Here's some recycled capacitors and a bridge rectifier being used to investigate converting an old dynamo torch to run white LED's instead of a bulb.