What is the function of power supply connector?

On a desk top, it connects power from the outlet on the wall to the power supply in the computer.

On a laptop. It converts the mains AC power at high voltage to a low voltage DC. Most laptops use around 19v DC.

This is then used to charge the battery in the laptop. When the battery is full, the laptop is then powered by the mains supply, leaving the battery in standby.

On a desk top the power supply is inside the cabinet. On a laptop it is separate. This saves weight and space when using the laptop totally portable and running on battery