The Basic Idea
The basic idea of an e-bomb -- or more broadly, an electromagnetic pulse (EMP) weapon -- is pretty simple. These sorts of weapons are designed to overwhelm electrical circuitry with an intense electromagnetic field.
If you've read How Radio Works or How Electromagnets Work, then you know an electromagnetic field in itself is nothing special. The radio signals that transmit AM, FM, television and cell phone calls are all electromagnetic energy, as is ordinary light, microwaves and x-rays.
For our purposes, the most important thing to understand about electromagnetism is that electric current generates magnetic fields and changing magnetic fields can induce electric current. This page from How Radio Works explains that a simple radio transmitter generates a magnetic field by fluctuating electrical current in a circuit. This magnetic field, in turn, can induce an electrical current in another conductor, such as a radio receiver antenna. If the fluctuating electrical signal represents particular information, the receiver can decode it.
A low intensity radio transmission only induces sufficient electrical current to pass on a signal to a receiver. But if you greatly increased the intensity of the signal (the magnetic field), it would induce a much larger electrical current. A big enough current would fry the semiconductor components in the radio, disintegrating it beyond repair.
Picking up a new radio would be the least of your concerns, of course. The intense fluctuating magnetic field could induce a massive current in just about any other electrically conductive object -- for example phone lines, power lines and even metal pipes. These unintentional antennas would pass the current spike on to any other electrical components down the line (say, a network of computers hooked up to phone lines). A big enough surge could burn out semiconductor devices, melt wiring, fry batteries and even explode transformers.
There are a number of possible ways of generating and "delivering" such a magnetic field. In the next section, we'll look at a few possible EMP weaponry concepts.