The Pickup Primer

The Electric guitar is different from an acoustic or classical guitar in that it has one or more electromagnetic pickups installed on it. These convert the mechanical to and frow movement of the metal strings (usually nickle) into an electrical signal that can be amplified. They consist basically of a coil of very light gauge wire around metal posts resting on a powerful magnet.
The string movement produces a small alternating current which is reproduced in an in and out fashion by the speaker. As the speaker moves it causes vibrations in the air and 'voilà', sound is produced.

The more winding there is, the more current is produced by the pickup and the warmer your sound will be. If you like your guitar to produce a brighter sound, get a pickup with less winding, this will also produce less power. The choice of pickup is entirely up to you as to the sound you want to achieve.

Single Coil or Double Coil (Humbucker)
A single coil pickup consists of only one row of metal posts with a single winding around them, whereas a double coil pickup or humbucker consists of two rows of metal posts and a coil around each row.
These are usually hooked up in a way that the center tap of one of the coils is connected to the outer tap of the other. This cancels the hum or buzz which is frequently produced by single coil pickups especially at high gain.

A double coil pickup is 'two pickups in one'. Since they are side by side, besides having double the winding, they catch more string area. This results in a fatter more powerful sound.
If you're in a working band which does a wide array of musical genres, you may want to get the best of both worlds by getting a single coil tap switch installed on your instrument. You'll have to also adjust your effects pedals or amp to compensate the volume when you flick the switch as the double coil pickup produces twice the power of a single coil pickup and you'll definitely notice it.

The Piezo Electric Pickup

This type of pickup is a practical invention especially for live performances with acoustic instruments because the musician is not 'tied' to a mic while playing his part and has the freedom to move around the stage.

You can find Piezzo Electric Pickups on acoustic electric instruments or you can get them installed in a guitar, voilin, mandolin, stand up bass, etc. They work like a condenser microphone and consist of two metallic plates separated by a dielectric material. Being that they don't give out as much power as an electromagnetic pickup, they need to have a special preamp installed in the instrument, which tapers the sound and amplifies the signal enough to get through 10 or 20 feet of cable to the amplifier.

They pick up any vibration from the instrument: string movement, knocks on the instrument -even someone talking into the sound hole. On acoustic string instruments they are installed under the bridge -it's important to note that they must be well stuck to the under side of the bridge or you won't get a very good sound from them. A good luthier knows this and I would highly recommend handing your instrument over to one to do this job. Being that they work microphonically, these instruments tend to feedback easily if they are turned up too loud or improperly equalized.

This type of pickup is good for acoustic instruments because they are better for getting the natural sound of the instrument and the strings don't have to be metallic. All this said though, if you're doing serious studio recording, I recommend using a good mic intended for the job and finding the best placement for it. You'll find that it just sounds better.

You can also check out the assortment of piezzo electric and other pickups for acoustic instruments at zZounds.com