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Types of speaker drivers

Electrodynamic Cone

Electrodynamic Dome

Electrostatic

AMT

Plasma

Piezo

Ribbon:  A thin ribbon made of aluminum is placed in a magnetic field between two fixed magnets of opposite polarity.  The ribbon acts as both the voice coil and the diaphragm, moving in the magnetic field in response to changes in the electric current running through it.    Because the diaphragm is almost as lightweight as the air around it it can produce extremely high frequencies with virtually no resonance whatsoever.  It moves very little air so it needs to be horn loaded to achieve an efficiency high enough for a larger room.  Good ones are extremely expensive and high end nearfield speakers/monitors will use them without horns.  Making them great for transparent music listening at close range.  The wide horizontal dispersion and tight vertical dispersion make it great for off axis listening and great for carrying sound long distances when stacked.

Planar-Magnetic:  Less expensive than a ribbon tweeter.  Frequency range isn't as high and it has more resonance.  The diaphragm is heavier, it's a piece of plastic with a voice coil wrapped around it.

Horn Loaded:  Any other type of driver behind a horn or flair which increases efficiency by controlling the dispersion pattern of the sound waves.  The driver at the base of the horn is called a compression driver.

Parts of an electrodynamic cone/dome

1.  Diaphragm:  As it moves it increases/decreases the pressure of the air inside/on top of it, creating sound waves.  This should be stiff, strong, and light weight.  Since it is always heavier than the air around it it always creates some type of resonance which is commonly reffed to as "coloration".
2.  Frame/basket/Chassis:  The "case" that hold everything together.  Can be closed off or open.
3.  Surround:  A flexible material (usually rubber) that connects the edge of the diaphragm to the frame. Allows excursion (in and out movement) of the diaphragm while the diaphragm remains stiff and doesn't change it's shape.  Sometimes a second surround is present at the center of the diaphragm to connect the diaphragm to the spider or voice coil.
4.  Dust cap or phase plug:  Used to fill the gap between the diaphragm and the pole piece.  Both prevent dust from getting in and reduce cavity resonances that occur inside that space by filling it.  Phase plugs are more expensive but do a better at promoting uniform motion and reducing resonance (it can also help boost bass by giving the diaphragm more momentum and help control dispersion patterns).  The dust cap is a plastic hemisphere and a phase plug is a bullet shaped plug made of metal or wood.
5.  Pole piece/top piece:  Holds the fixed magnet in place.  Pole piece is underneath, top piece is on top, and the fixed magnet is sandwiched between them,
6.  Voice coil:  A coil of copper wire wrapped around a hollow cylinder shaped piece of metal and attached to the frame by the spider.
7.  Fixed Magnet:  A circular magnet with a hole in the middle of it that provides a fixed magnetic field for the electromagnet's magnetic field to interact with.
8.  Heatsink:  A piece of metal surround the magnet or pole piece that absorbs heat.
9.  Ferrofluid:  A magnetically suspending coolant applied to the voice cool to prevent overheating.
10.  Spider:  A suspension system connected to the frame at the edge, the base of the diaphragm on one side of the center and the voice coil on the other side.
11.  Tags:  Connected to the voice coil on one side and the +/- wires carrying the analog audio signal on the other side.  Provides a convenient way to wire the driver without screwing with the voice coil.
12.  Magnetic shielding:  Provides one or two way protection from internal or external magnetic force.

The diaphragm is usually manufactured with a cone- or dome-shaped profile. A variety of different materials may be used, but the most common are paper, plastic, and metal. The ideal material would be 1) rigid, to prevent uncontrolled cone motions; 2) have low mass, to minimize starting force requirements and energy storage issues; 3) be well damped (absorb acoustic waves), to reduce vibrations continuing after the signal has stopped with little or no audible ringing due to its resonance frequency as determined by its usage. In practice, all three of these criteria cannot be met simultaneously using existing materials; thus, driver design involves trade-offs. For example, paper is light and typically well damped, but is not stiff; metal may be stiff and light, but it usually has poor damping; plastic can be light, but typically, the stiffer it is made, the poorer the damping. As a result, many cones are made of some sort of composite material. For example, a cone might be made of cellulose paper, into which some carbon fiber, Kevlar, glass, hemp or bamboo fibers have been added; or it might use a honeycomb sandwich construction; or a coating might be applied to it so as to provide additional stiffening or damping.

Smaller drivers move less air and therefore can't produce low frequencies very well but are faster and lighter allowing them to produce higher frequencies better. 

Domes offer a wider dispersion and are better at producing high frequencies but don't move as much air as cones and therefore can't produce low frequencies as well as cones. 

Lower frequencies have wider dispersion patterns than higher frequencies and require more air to be moved (per wave) to achieve the same volume as a higher frequency. 

Higher frequencies are more susceptible to resonance but with lower frequencies uniform motion is more of an issue.  With all this in mind you can see why midrange drivers are medium/small cones, tweeters are small domes, and woofers are large cones. 

Typically a two way speaker uses a short throw cone woofer and a three way speaker uses a long throw cone woofer with a short throw cone midrange.  Short throw means the curvature of the cone is very shallow and long throw means the curvature is very steep (therefore the cone is deeper and traps more air).  Long throw cones trap more air and are therefore better at producing low frequencies but in exchange are not as good at producing midrange/high frequencies as short throw cones since they are bigger and deeper.  Subwoofers use very large long throw cones.