When I press the on button of my PC, it starts. But nothing comes in my monitor. I opened my casing & found that my fan doesnt rotate freely. It takes sudden breaks while rotating. After some time the fan stops completely & the PC turns off. One of the push pins around the fan seems loose. My motherboard is of Gigabyte. I am still to figure out the problem & dont have any idea of what to do.

If you speak of the CPU fan, which appears you do, then the "loose pin" would cause a lost contact between the heatsink and the CPU thus causing extreme overheating which would force a shutdown for protective reasons.

You need to re seat that heatsink/fan.

If the pin is broken you need a new heatsink/fan.

If the motherboard "hole" is broken you will have to find an alternative mounting configuration short of replacing the motherboard.