It's a smoothing capacitor, your motherboard will function ok for the most part without it. A large capacitor like this is normally used for pre regulation filtering, to smooth out ripple on the power line. Longer term, excessive ripple causes strain on the regulation circuitry as it has to work harder to provide the power/low impedance at the noise related frequency. However with pc's products are seldom used long enough for the failure to surface.
Capacitors you don't want to knock off are those used for compensation in feedback networks, they are small smt types usually in the pico farad region, there are many smt capacitors on motherboards and graphics cards, though few of them are absolutley critical for operation, most provide high frequency filtering to scotch rfi/emi or switching noise.
Based upon your pictures, if you wish to replace the capacitor replace the capacitor with a new one of equivalent voltage and micro farads. Or if unsure leave as is. BTW reason I suggest using a new capcaitor instead of the old is if you have pulled on the terminals, there is a chance they have shorted and will cause component damage upon reuse, better to be safe..
Not all damage to capacitors is always a catastrophe.
regards
raja
