How do you test the output of a household AC electrical power adaptor?

First, always read the label on the power adaptor to find out what voltage it has been designed to supply.

You can use a volt meter or a 12v light bulb to check if the power adaptor is working but don't do this unless you are certain that the output voltage really is a low as that!

If you don't check the label first you could damage the volt meter or the bulb and/or give yourself a dangerous shock.

Another more technical answerFirstly, using a voltmeter to check that the output voltage is as specified on the label. An unregulated power adapter will usually have an output voltage somewhat higher than the voltage specified when measured using the meter alone, but will drop to nearer the required voltage when measured under a significant load. A regulated power adapter should give a fairly constant voltage reading when tested with or without a load applied. Second, using an oscilloscope to see if there is any significant "ripple" in the output voltage, the presence of which would indicate a faulty smoothing capacitor inside the adapter.