Get your motor runnin'
Assuming you are already familiar with bicycles
Let's just focus on the parts that are different from a standard bicycle, shall we? In the hub of the front or back wheel of an electric bicycle you'll find a hub motor. Inside the hub is the standard story of an electric motor: magnets, coiled wires, armatures, brushes (and sometimes not). However, in a typical motor, the axel is the thing that moves and drives some machine. In a hub motor, the axel is held stationary; meaning the rest of the motor must spin. This results in a smaller, more efficient means of locomotion than would be achieved with a regular electric motor. Whether the motor is powered by throttle or some other mechanism really doesn't alter the operation and is really a matter of preference.
Since these electric motors can get quite powerful, the wheel attached to it must have much stronger spokes. Likewise for the fork (if it is a front wheel motor) and frame, the greater the torque of the motor, the stronger the materials must be. This creates something of a paradox, because the materials must also be light to achieve the greatest range.
A few words on batteries
Sealed lead acid batteries are the most commonly used batteries for electric bicycles because of their price and ease of use. Everybody who has owned a car knows this kind of battery is reliable. They are, however, the heaviest and cannot be quickly charged.
Nickel metal hydride batteries are high-energy-density, quick-charging batteries and are considered the next "step-up" in technology. They have a higher leak rate than most batteries, meaning you can't leave them lying around for a long time and expect a full charge.
Lithium-ion batteries are an excellent choice if you need quick charges and top performance. They require some accompanying technology to maintain- a battery management system to maintain stability. Naturally, this type of battery is getting into the higher price levels.
Lithium-polymer batteries are the lightest on the market and represent the latest technological developments in batteries. At around 12 pounds, they are less than half the weight of sealed lead acid batteries and just as stable. Currently, prices for these batteries are around 600 dollars.