So my problem is that my motherboard has enough space for the 24 pin motherboard power connector. However when I have plugged in all the 24 pins, my computer has failed to start. So then I reopened my case and took out the small connector that has 4 pins, which evidently lead to the motherboard responding. Why did this problem occur and are there any serious implications it may lead toward my computer?
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You seem to be powering a Mobo equipped with a 24-pin power connector from a power supply that has a 20-pin power connector and a 4-pin power connector, where the 4-pin-connector is the "auxiliary" type that provides GND and +12V (two black, two yellow wires) and not the "20+4"-type that has the wires for pins 11, 12, 23 and 24 of the 24-pin receptacle on the MOBO (+12V/yellow, +3V3/orange, +5V/red and GND/black). Most power supplies have the "aux" type, only some have the "20+4" combination needed for compatibility with both 20-pin Mobo connectors and 24-pin Mobo connectors.
The pinout of the 24-pin connector can be found at wikipedia.
From your question, it sounds like you are connecting pins 1, 2, 3 and 4 of the 4-pin-connector from the power supply to pins 11, 12, 23 and 24 of the motherboard, thereby shorting Ground to +3V3 and +12V via pins 11, 12 and 24 on the motherboard connector and +5V to +12V via pin 23 on the motherboard connector, all in all, this is a short connection of Ground, +3V3, +5V and +12.
You are kind of lucky nothing got blown.
You might be fine if you just use the power supply's 20-pin connector for pins 1-10 and 13-22 on the motherboard because pin 11 is parallel to pin 10; 12 is likely parallel to 1,2 and 13; 23 is parallel to 3, 6, 21 and 22; and 24 is parallel to 3, 5, 7, 15, 17, 18 and 19.
No guarantee though that there is a bit more current than specified through the wires and pins because for the affected nets (+3V3, +5V, +12V and GND), there is less copper (=wires) available as intended by the Mobo manufacturer.
Do not connect the small 4-pin connector (2 yellow, 2 black) onto anything than a matching 4-pin-connector on the Mobo. If the Mobo does not have a 4-pin power connector, do not try to connect the 4-pin connector that comes from the supply to any point.