But First, the Complicated Bit
I'm going to keep this as simple as I can, but there are one or two things you should know before we start.
A modern (less than 10 years old) computer's power supply unit (PSU) is designed to supply five voltages at varying power capacities. The voltages are +3.3V, +5V, -5V, +12V and -12V. Because the computer has different power requirements at each voltage, the PSU design is optimised to provide the appropriate power to each rail. A rail is basically a single voltage, so there will be a +3.3V rail, a -5V rail, a +5V rail and so on. If you count the rails you'll see it doesn't match the number of wires exiting the PSU. Don't worry about that - it's just for technical convenience.
As technology and miniaturisation plodded on, so the operating voltage of the components reduced (smaller components require less voltage). As more and more components became lower voltage, the power requirement at that lower voltage increased. Without getting too technical, it became more efficient and practical for the motherboard manufacturers to convert the 12V rails to the actual voltages required at the component locations on motherboard. Transporting the power around the motherboard at this higher voltage (and, therefore, lower current as we'll see in a moment) enabled the motherboard conductors to be narrower (roughly an eighth of the width) and, therefore, more densely packed. The reason for this is to be found in the power calculation:
P (power in Watts) = I (current in Amps) x V (Volts)
for the purposes of this explanation, I'll turn that round to give us the current (I) as the result of the equation:
I = P / V
e.g. at 3.3V, a CPU requiring 65W will draw a current of 65/3.3 = 19.69 Amps. Thats almost 20 Amps going through those thin conductors!
At 12V, however, the current will be a much more manageable 65/12 = 5.41 Amps.
Okay, so how does this affect you? Well, older PSUs of the same wattage as newer ones may have a much lower power rating on the 12V rails. It would not be a good idea, therefore, to cannibalise an old PC for its otherwise perfectly serviceable PSU to use in a newer PC. The chances are, you would be overloading and overheating it to some extent. Conversely, a new PSU used to repair an older computer may not supply sufficient power at the lower voltages, again risking overloading. More important here is that the -5V rail is no longer used on motherboards, so new PSUs tend not to provide a -5V rail. If you're using a PC that old, you'd be better off replacing the motherboard and processor, but that's another story.
Now we've got that out of the way, let's take a look at how to determine the wattage of the PSU we need.