In more technical terms, this is is what actually happens to the capacitor... Think of an electrolytic capacitors as a battery. They are designed to store a charge and release that charge depending on the specific requirements of the circuit. Inside the capacitor there are two metal plates with dielectric material between them, wrapped in paper, filled with acid (electrolyte), and sealed in its housing or 'canister'. What happens is the flawed electrolyte prematurely deteriorates and dries up. When this happens the capacitance value changes, becomes erratic, and can even short completely, which obviously causes the circuit to malfunction. On your motherboard, this results in system instabilities or complete failure of your board.
From a physical standpoint, the capacitor can display a number of symptoms and even have catastrophic failures. Catastrophic failure is a rare phenomenon, but it does happen. The reason is this... A capacitor 'canister' is completely sealed and air tight. When the flawed electrolyte dries it turns from a liquid state into a gas. This gas expands with heat and builds great pressure inside the canister, the theory is the same as a pressure cooker. Of course the obvious happens when that pressure builds too much. This is what causes the capacitor to 'bulge' or swell up. In a catastrophic failure, the capacitor may actually burst or explode. It can sound like a firecracker going off or sound similar to air escaping from a car tire, depending on how high the pressure has built. A physically failing capacitor has an ammonia-like odor (at least that what I thought it smelled like)... Capacitors that has swollen up are easy to detect, but one that has burst are even more noticeable. It will usually blow from the top and spew its innards throughout your computer case. The innards are mostly paper and any remaining electrolyte, but it sure can leave a mess... Once again, catastrophic failure is a rare phenomenon... It's not dangerous either, as long as you practice common sense safety precautions! If you suspect bad caps, don't put your face near them while the system is powered up, and don't eat the paper or electrolyte that they blow out!