Lets try to understand the importance of the pick-up in the chain that generates the final sound of the electric guitar. The transducer, which is an electronic or electromagnetic device capable of converting physical energy into electrical energy, is well adapted to the pick-up. The pick-up, in fact, has the job of converting the energy produced by the vibration of the strings into alternating current (AC) electrical impulses, that are directed to the amplifier, which manipulates the signal that is then sent to the speakers. The pick-up of an electric guitar is usually magnetic, in the sense that it has a magnet that produces a magnetic field in which the strings pass through. In fact, the pick-up only works when it is near the strings, and are therefore mounted directly beneath them.
There are two basic catagories of pick-ups: the single-coil and the humbucking, with their variations (active and splitable pick-up's).

The most basic scheme of a pick-up has a fixed bar magnet that is wrapped with very thin copper wire (about the size of a human hair). The wire is wrapped around the bar many thousands of times, creating an electrical coil. The magnet generates a magnetic field around the coil and the strings of the guitar, and when the pick-up is mounted just below, it passes through this field, interacting with it. When the string is stopped, the magnetic field is inert. When the string vibrates, the form of the field is altered.

The magnetic bar inside the pick-up. The copper wire wrapped around the central portion of the bar.

The alteration of the force lines within the magnetic field comes from small impulses of electric energy, which then go to the amplifier in the form of alternating current. The vibration of the string is different for every note played. For example, the A note if tuned correctly, will vibrate at 440 cycles/second (440 hertz). Looking at the figure below and left, the vibration will cover the figure eight 440 times. The magnetic field that surrounds the pick-up will therefore be altered 440 times.

Slowly as the volume of the note fades, so does the distance between points -1, 0 and +1 (figure eight above right), until the 3 points are perfectly coinciding at the 0, which is when the string is practically motionless. This however does not vary the actual number of cycles/second, which remain invaried until the end of the vibration. The number of cycles/second also indicates the tone of the note. The figure eight that we see graphically defines the tone of the note.
The materials used to make the magnets of the pick-ups can be alnico, nickel, and cobalt. Some manufacturers use ceramic or piezo (a crystal) magnets. The shapes of the magnets are different depending on the case. In fact, there are bar magnets that have six individual polar expansions, whose heights can each be modified in order to be more or less close to the strings with respect to the others. In other cases we have the blade magnet, which is a central blade that produces a constant magnetic field over its entire length.
The number of times the copper wire is wrapped around the magnet greatly influences the sound that the pick-up is able to give. Generally, the power of the pick-up increases in relation to the type of winding, even if you need to limit the number of turns, which could transform the coil into a type of magnet that stops the vibration of the strings. A coil could have from 7,000 to about 8,500 turns.

The single-coil pick-up

The single coil pick-up was introduced for the first time on the electric guitar of one of the most ingenious and innovative person in the field, Leo Fender. His research, aimed mostly at electronics (he was owner of a radio repair shop), created the Broadcaster, then the Telecaster and finally the Stratocaster, whose sound characteristics came from the precise choice of woods as well as the single coil. The Telecaster has two pick-up's, while the Stratocaster has three.

The typical configuration of a Stratocaster requires the use of three single-coil pick-ups, that are defined according to their position (the neck, center, and bridge pick-ups) on the body of the instrument. A three or five position selector lets you choose one of these, or the neck and center together, or the bridge and center together.
One volume control manages all the pick-ups, while the neck and center pick-ups each have a tone control.

The model of pick-up seen on the side is what is mounted as standard on the Fender Stratocaster. This single-coil pick-up is present on numerous models of different marks of guitars. It's main characteristic is brilliant sound with many high pitches. In the diagram, we can see how the six polar expansions are wrapped around the bobbin of copper wire, where the two ends of the coil are connected to two cables connected to the electrical circuit of the guitar.

Single-coil
Stratocaster

Single-coil
blade

Single-coil
Telecaster at the neck

Single-coil
Telecaster at the bridge

Il pick-up humbucking

The twin reversed pick-up is usually called "humbucking", because its a compensator for buzzing or humming and is void of background noices. A single-coil pick-up, with its pure tone, has one small defect: it has a high risk of background noise. This is due to the fact that all pick-up bobbins are sensitive to interference coming from electromagnetic radiation, in that they tend to gather noices from electric devices, amplifiers, radio waves, etc. In 1955, a pick-up was designed by Seth Lover, a technician for Gibson, that neutralized these noises (called "humbucking). The innovation was mostly in the fact that there were two bobbins, connected in series. Initially the current passes as a positive signal through on bobbin and then through the other as a negative signal. The two currents, traveling through the bobbins in opposite directions, cancel each other out: in this way any eventual humming would not be transmitted to the amplifier. But the signal generated by the vibration of the strings in the magnetic field must get to the amplifier. This is why the series of poles on each bobbin have opposite magnetic polarities. When the secondary bobbin inverts, the signa from the magnetic field duplicates, instead of cancelling, the electric impulse. When the two signals vibrate in the same ways, they are called "in phase", and if they are opposite they are "out of phase".
The sound quality of the pick-up is quite different from that of a single-coil: the humbucking has a more full sound and a lower definition of the higher frequencies. For this reason there are two types of sounds: the stato sound, characterized by the use of the single-coil pick-up, and the Gibson sound with the typical humbucking sound.
The dimensions of a humbucking pick-up, being a double bobbin, are practically double with respect to the common single-coil. In many cases there has been requests to mount the humbucking on guitars that origionally had single-coil, an operation that can be done by just modifying the hole in the wood where the pick-up is to be inserted. But for those who don't want to touch the body of their instrument? Some companies, such as DiMarzio and Seymour Duncan, have overlapping (or side-by-side) double bobbin models in single-coil format.

In the figure on the side you can see the structure of a double bobbin pick-up on an origional model Gibson, a model thats often been imitated and varied. The main bobbin is equipped with adjustable expansions, while the secondary bobbin remains fixed. The series of poles are in contact with the magnet placed under the bobbins. In some models, like the Gibson PAF humbucker, the pick-ups are enclosed in a metal box. For the bobbin, the number of turns of the copper wire can vary depending on the choice of tone destined for that pick-up.

Humbucking
PAF Gibson

Humbucking
Seymour Duncan
in single-coil format

Humbucking
Gibson

The splittable pick-up

The possibility to divide the bobbins of a humbucking pick-up in two has always been the source of an increase of the sound variety of the pick-up itself. This system is therefore used to give more sounds to the instrument that has a humbucking, and consists of a selector switch thats placed between the pick-up and the controls of the guitar. With this switch, you have the possibility of using either one of the bobbins of the pick-up, or both.

The passive pick-up

This term identifies the models of pick-ups analyzed up to now. It refers to the fact that the generation of electric signal by the pick-up is clean, in the sense that there are no reinforcements or controls that can alter the signal (apart from the classical tone and volume controls, that are always present). The body of the sound in a passive pick-up is therefore natural.

The active pick-up

The sound of an active pick-up varies according to the construction method. The most famous brand for the guitar is surely EMG, which has many different models, both single-coil and humbucking, that are equipped with dedicated preamplification circuits. The most evident qualities of these type of pick-ups are the high dynamic yield and the clarity of the signal that they are able to transmit to the amplification (rule = absence of background noice, typical of a passive pick-up). Many guitar manufacturers mount active pick-ups as standard equipment. An active pick-up, furthermore, has a modern sound: this is why the purists of vintage sound shy away from them. In my opinion, its best to have the possibility to play guitars with both active and passive pick-ups: each are very capable of performing their job. Finally, a few words regarding active preamplification circuits. In fact, its possible to combine an electronic circuit to passive pick-ups that can preamplify the signal of the pick-up, enhancing it through a series of tone and frequency controls. The purpose of this type of circuit is to increase the sound possibilities of the pick-up itself.