Put the power supply under a load and measure voltages with a multimeter. The most meaningful load test is to have it power the whole system as you've assembled it. However, that potentially puts the hardware at risk if the PSU is bad.
Since it's not even powering on, you could disconnect the motherboard and just have it power a disk drive. This avoids risking the motherboard and might make it more convenient to test. Just be aware that if the PSU is very lightly loaded, it's voltages might not be in spec until the load is increased.
Make sure the motherboard power connectors are in the right places as Deksor mentioned.
You can measure the +5V and +12V at the 4pin molex drive connectors.
Put the black probe to ground - easy places for ground are against the PSU casing or stick it into one of the terminals with a black wire.
Put the red probe into a terminal with the red wire to measure +5V, and the yellow wire to measure +12V. Don't put the black and red probes into the same molex connector, because if they touch you might cause a short circuit.
-5V and -12V can only be found on the motherboard connectors. I'm not sure which wires they are, but I think one of them is the white wire.
You may be able to get the red probe into the backside of the connector and get a measurement while it's plugged into the board.