Definition
A power surge is a period of abnormally high voltage in an electrical system, which pushes too much current though electrical devices. This can trip circuit protection devices or even damage electrical equipment. Surge protectors, such as a power strip, can protect your equipment from these damaging effects. A very brief power surge that lasts less than one power cycle is called a voltage spike.
Effects
Since a power surge manifests itself as excessive voltage, and the current flowing through a device is equal to the voltage applied divided by the resistance of the device (which is constant), more voltage → more current. More current means more heat, and too much heat can melt insulation, causing short circuits, or fry delicate transistors, or even cause something to catch fire. Even if these things don't happen, the extra strain this places on the equipment can reduce its operational life. Very high power surges can go so far as to trip circuit protection devices, but it is uncommon for them to get so high that a standard household 15 or 20 Amp circuit breaker would trip. Higher-quality electronics may have protective fuses built-in which should trip before these currents reach damaging levels, but cheaper electronics and appliances will not.
Wall voltage in the US is regulated to stay between 110 VAC and 120 VAC under normal circumstances. A surge of 130 VAC can significantly shorten the operational life of an incandescent light bulb. A surge of 140 VAC can blow it out almost immediately.
Causes
Various things can cause a power surge. One disadvantage of our modern power generation and transmission system is that the large, centralized electrical generators are slow to react to changes in demand. Fortunately due to the wide distribution of demand among millions of homes and businesses, they normally average out to something manageable. Catastrophic events on the grid, however, can cause sudden large changes that the generators can't react fast enough to compensate for. A massive blackout, for example, can suddenly reduce the load on the grid by cutting off entire cities. Until the generators can reduce their output, they're pumping excessive wattage out into the grid, which drives up the voltage, resulting in a power surge.
The opposite situation, in which demand remains constant but a power generator suddenly goes off-line, can create a voltage sag until the other generators can increase their output to compensate. Likewise, a very large factory suddenly starting up or shutting down can cause a localized sag or surge by quickly changing the demand.
Lightning strikes can also cause local power surges, or voltage spikes to be more accurate, due to their brief nature. Lightning, being of course a giant electrical spark, can either directly pump more electricity into the grid with a nearby strike, or induce it by creating an electric field.
An electromagnetic pulse is another means of generating a large electric field that can induce a power surge on the electrical grid. An EMP can be a side-effect of a nuclear explosion, or it can be created directly by a dedicated EMP bomb, although these are only theoretical now.
Finally, a coronal mass ejection is a cloud of charged particles thrown out of the sun by a solar flare. If one of these ejections happens to be aimed at the Earth, it can cause a power surge by inducing currents in the power grid. There isn't much that can be done about this, but early-warning satellites can spot them and give advance warning. Fortunately there is a 50% chance that this cloud of particles will have a magnetic field lined up the right way to bounce off the Earth's magnetic field. Unfortunately that leaves a 50% chance it won't.
Protection
There is very little than can be done pro-actively to prevent power surges, especially from the consumer's end. Fast-reacting, low-inertia power generation like hydroelectric dams can speed up the power grid's reaction time, but cannot eliminate it entirely.
From the consumer end, the best reactive options are surge protectors, such as a power strip, or an uninterruptible power supply. A surge protector usually takes the form of a metal-oxide varistor. This is a normally high-resistance circuit element that reacts to increases in applied voltage by reducing its resistance. By placing one of these across the hot and neutral conductors of a power strip, it has the effect of clipping the voltage sine wave before it can get abnormally high by shunting some of the power around to the neutral so it doesn't pass through the appliance plugged into the power strip. A power strip usually also has a fuse or a circuit breaker as an additional level of protection.
An uninterruptible power supply is a battery back-up system for things that normally plug into a wall. It charges the battery off of the wall voltage, and provides clean, stable power to the appliances through a DC to AC inverter. An active UPS always does this, and a passive UPS switches from letting the wall voltage go through unimpeded to powering from the inverter only when it detects abnormalities in the wall voltage. These have a longer operational life but can react more slowly to disturbances.
As more and more appliances are powered from DC, however, this is becoming less of an issue. Many modern AC to DC converters are very tolerant of power quality issues, able to convert large voltage ranges and fluctuations in input frequency while still maintaining a stable DC output voltage, within reason of course. Large voltage spikes can still overpower them and damage the equipment they are plugged into.