you can plug a 20 pin cable into a 24 pin motherboard and it will work just fine. At least, it will for a while. They added those extra 4 pins for a reason. When you plug a 20 pin cable into a 24 pin connector you're not providing the extra current carrying capacity which may be needed by the motherboard. If your motherboard's current requirements are low enough then it will work properly with only a 20 pin power cabled plugged in. But if the motherboard draws enough current, then you can overheat the 20 pins you're using on the 24 pin connector. I've seen enough pictures of burned ATX main connectors to assure you that this happens. In adding the extra four pins, 3.3 volts went from 3 to 4 lines. 5 volts when from 4 to 5 lines. 12 volts went from 1 to 2 lines so its capacity doubled. Connectors really do get hot if you overload them so the safest thing to do is use a real 24 pin power supply on a motherboard with a 24 pin connector. Note that your 24 pin machine may work fine with a 20 pin power supply until you add a PCI-Express card later on down the road. PCI-Express cards can draw up to 75 watts so they can cause problems. Many PCI-Express video cards which have the 6 pin PCI-Express power cable still draw a substantial portion of their 12 volt load through the PCI-Express slot. The extra 4 pins doubled the current capacity of the 12 volt rail so that one is easy to overload when only using a 20 pin main power cable. A 20 pin power cable only fits into one end of a 24 pin motherboard connector so you can't insert it incorrectly.
The picture on the right shows an adapter which allows you to plug a 20 pin power supply into a 24 pin motherboard. These do not solve the problem. In fact, they usually make things worse. You still have only the same old 20 wires and pin&socket connectors that you would have if you plugged it directly into the motherboard. But with the adapter you also have longer wires and another set of pins&sockets so you have a higher voltage drop and more things which can go wrong. All the adapter does is remove the problem from the 24 pin motherboard connector and move it to the 20 pin connector on the adapter. About the only good thing it does is to make sure the 20 pin connector on the adapter melts rather than the 24 pin connector on the motherboard. You shouldn't underestimate the potential problems which come from adding another connector between the power supply and the motherboard. The resistance of connectors can increase due to being plugged and unplugged. Increasing the resistance makes the connector drop more voltage and dissipate more heat. Any number of physical changes can also cause higher connector resistance which causes more problems. You shouldn't put an extra connector in a high current path unless you've got a good reason. That's a good rule for power cables in general but it goes doubly for the high current draws you sometimes find through motherboard main power cables. The best practice is to avoid these 20 to 24 pin adapters and get a real 24 pin power supply.
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