The first electric guitars all used single-coil pickups. As you might expect, a single coil pickup has only one coil of wire. It may have -

	a single magnet
	a single magnet with screws for adjustable pole pieces
	a separate magnet for each string

Regardless of the number and arrangement of magnets, it is still a single coil pickup if it has only one coil of wire.

Unfortunately, in addition to producing an electrical signal from a vibrating magnetic field, a coil of wire has another less desirable (for our purposes) property - it is a very efficient antenna. A coil of wire will "pluck" electromagnetic radiation out of the air, and we are surrounded everywhere by this radiation - most notably the 50/60 HZ hum from building wiring, nasty electrical noises from fluorescent lighting, and so on. Single-coil pickups are susceptible to hum. The hum can be controlled to a great degree with shielding and careful wiring - but it seems many guitar manufacturer's can't be bothered with this.

In 1955, a Gibson engineer by the name of Seth Lover invented a new type of pickup he called the "humbucker." The technical name for this technique is hum-cancelling and I am sure that the Gibson company management looks back now with 20/20 hindsight and wishes dearly that they'd protected the term "humbucker" with a registered trademark. A humbucker uses two coils and either two magnets (or sets of magnets), or pole pieces at opposite ends of a single magnet. Contrary to popular belief  it is incorrect, or at least very misleading, to say that the two coils are "out-of-phase."

When speaking strictly of electrical coils by themselves, they are said to be "in-phase" when they are wound the same direction. However, pickups are said to be "in-phase", when their signals are in phase (the signal generated in one pickup adds to the signal generated in the other, instead of subtracting from it).

The polarity of a signal generated in a conductor depends on the magnetic polarity of the flux lines and the direction of travel. In a humbucking pickup, the two coils are wound with opposing electrical polarity, but the magnetic polarity for each coil is also reversed. Therefore, in a humbucking pickup we have two coils of wire, each of which is carrying two electrical signals. Each coil has the intentional signal generated by string vibration within the magnetic field and the unwanted noise signal picked up from ambient electromagnetic radiation. The polarity of the desired signal is dependent on both the polarity of the magnet and the direction of the coil winding so this signal is "in-phase" across the two coils. The polarity of the noise signal is independent of the magnets -- it depends only on the direction of the coil winding - so the noise signal across the two coils is cancelling, or out-of- phase.

Tapped coils and split humbuckers

Pickup selection can get quite confusing, especially with the huge variety of pickups available now from Seymour Duncan and others. Now, we not only can select from single-coil and humbucker pickups; there are tapped-single-coils, stacked humbuckers, and four-wire humbuckers (which may or may not be stacked):

Tapped single-coils are possibly the most confusing, as many people mistakenly think they are hum-cancelling. That is not the case. They are in every respect a single coil pickup. The only difference is that there is a third wire which leads to a "tap" partway down the coil. You can install a switch to short this tap to either end of the coil -- thus altering both the tonal response of the pickup and how strong a signal it generates. In theory, you could switch between a heavy-metal powerhouse and a sweet bell tone on the same pickup. In practice, it rarely works out quite that well. There are few tapped single-coil pickups, I don't know of any in current production.
Stacked humbuckers look like a single coil pickup, and will fit neatly in place of a stock single-coil, but they are actually humbuckers where the coils are stacked one above the other. Other single-coil sized humbuckers are sometimes called stacked even though technically they have coils side by side.
Four-wire humbuckers are simply humbuckers which have both ends of both coils brought out on the cable. Thus, you can use a switch to control how the pickup is wired, by switching between series/parallel wiring for the two coils or using either coil of the pickup as a single coil, etc. This is another case where practice doesn't quite live up to theory. Coil splitting is okay, but it takes a strong humbucker to really sound good split.

The Stratocaster middle pickup as a "humbucker"

On newer Strats (shipped since the mid-'80s -- except maybe some "vintage" reissues), the middle pickup has reverse magnetic polarity from the neck and bridge coils which is why selecting the second or fourth position is much quieter -- in essence the two coils work the way a humbucker does. Note that you can not simply reverse the leads of the middle pickup on older Strats (with north-north-north pickups) because, while the noise signal will cancel, so will the string generated signal! It is also important to know this bit of trivia when you are replacing pickups because both north-south-north and south-north- south Strats have been shipped. If you replace only one or two pickups, and replace them with ones having the "wrong" polarity, your axe is suddenly going to sound crummy!

Pickup design/selection issues

There are far too many variables in pickup design to even approach a thorough treatment of the subject here. However, the following tips will help you understand how pickups affect tone so that you can make a more intelligent decision if considering aftermarket pickups.

How "hot" the pickup will be depends primarily on two variables; how strong the magnet is, and how many turns of wire are on the coil. However, both of these factors also affect the tonal response of the pickup.
Any coil of wire is an inductor, and the impedance (total resistance) of an inductor varies with frequency (the higher the frequency, the higher the impedance). When you add more turns of wire to a coil, you are increasing the inductance and thus altering the frequency response. "Overwound" coils and humbuckers usually have less high frequency response (cleverly marketed as "stronger midrange").
Changing the guage (diameter) of the coil wire changes inductance and thus alters the frequency response of the pickup.
Changing the size or shape of the coil changes the inductance and thus alters the frequency response of the pickup.
Coils not only have a certain inductance, they also have a certain amount of "parasitic" capacitance. Different winding techniques will result in slightly different capacitance and thus affect the tonal response of the pickup.
Making the magnet stronger also has an effect on tone, in that the stronger magnet will generally give a sharper attack and "harder" tone. However, the greatest impediment to simply using very strong magnets to get a very high output is the fact that strong magnets will kill sustain by "dragging" the strings.
As a generalisaton -

	"Soft" magnet and fewer turns on coil - sweet, bell-like clear tones
	"Hard" magnet and fewer turns on coil - glassy hot Strat sound
	"Soft" magnet and more turns (or humbucker) - smooth, buttery midrange
	"Hard" magnet and more turns (or humbucker) - grunge or "Texas" sound.