An incredibly detailed and informative response there...
By 'power surge' people generally mean a voltage spike -- a short-term event where the normal supply voltage (120VAC in the US) goes above (potentially WAY above) normal. In the event of a lightning strike on a nearby power line (one of the more extreme events that can happen), the supply voltage might temporarily (for a few milliseconds, at least) be up in the hundreds of thousands of volts. The power supply in your computer is supposed to 'buffer' the internal rails from fluctuations in the external power lines. However, in such extreme circumstances, it often can't keep up, and the internal rails will often *also* spike (although not to quite such extreme voltages). By Ohm's law (V=IR), if you increase the voltage feeding a fixed-resistance circuit, the current (amps) flowing through the circuit will increase proportionally. This can be very damaging to the silicon logic gates used in computers, for complex subatomic physics reasons that I don't fully understand myself. Other components (such as voltage regulators, capacitors, etc.) also do not take kindly to such mistreatment. And if the voltage gets REALLY high, you can actually short across paths within chips, or even through PCB layers, physically destroying the circuits involved.