ESD, or electrostatic discharge, is the result of a difference in potential. ESD builds up through a positively (or less-negatively) charged object rubbing against a negatively (or less-positively) charged object. The positively charged object will pick up electrons from the negatively charged object, causing them to build up on the surface of said object in an attempt to equalize their charges.
ESD can indeed cause electronic failures, but these failures are of two types: hard failures, which involve a catastrophic destruction of a part of a component (eg, the dielectric of a capacitor, the gate of a MOSFET, or shorting through the P-N junction of a diode), and soft failures - a degradation of a part of an electronic component. An example of a soft failure with a diode would be if the discharge didn't short through the junction, but only increased the size of the junction by melting it a bit - entirely possible with the miniscule components found on circuit boards today. This would cause degraded performance of the component, and may never be noticed until the component entirely craps out.
ESD can be anywhere from a few volts to upwards of 35 kilovolts. If you can feel it, such as when taking laundry out of the dryer, the charge is at least 4kV. If you can see it, which sometimes happens when you zap yourself with a doorknob (try it in the dark), it's at least 5kV. If you can hear it, it's at least 10kV. To contrast, most electronic components will be killed by an ESD of 100 to 1000 volts, depending on the type of component.
One of the useful tools in determining how ESD can be built up is called the triboelectric series. I won't go into detail on it, but it's an example of substances that are positively and negatively charged. Basically, the farther apart they are on the list, the more charge they'll build up.
Finally, a great example of ESD that almost everyone has seen is a lightning strike. 75,000 volts and probably a couple hundred amps of pure electric zapping fun!