What is a Sealed, Ported, Bass Reflex, Acoustic Suspension, Bandpass, and Coupled Cavity Speaker? Which is better?

All are "direct radiator" enclosures, so called because the sound is produced directly from the driver (the "radiator") without the assistance of a contrivance such as a horn. To know the exact specifications for the construction of any enclosure, you must have the manufacturers list of the appropriate Theil Small Parameters for your woofer. Enclosures exist solely for the purpose of accommodating the woofer. Tweeters and midranges do not have to be enclosed at all. For more detailed instructions on speaker construction, see books at: Amazon.com

SEALED BOX: The simplest direct-radiator system. The rear of the driver sees a sealed enclosure, and none of the rear output of the driver contributes to the sound output. Depending on the ridgidity of the mechanical suspension versus how stiff the enclosed air in the enclosure is (and that's a function of the size of the box), you can have either an Infinite Baffle enclosure, in which the mechanical suspension is the dominant source of system stiffness and the box is large; or an Acoustic Suspension enclosures, where the air in the box is the dominating stiffness, and the box is small. Because energy must be expended in overcoming the internal pressure, sealed boxes tend to be the lowest efficiency systems for a given box size and bass cutoff frequency.

VENTED ENCLOSURES: Also the same as Bass Reflex, Ported, or Passive Radiator. Here, an aperture in the box provides a means for the rear output of the cone to contribute to the total output of the system. However, it only contributes over a very narrow range of frequencies. In fact, in a properly designed system, the front output of the cone is reduced at the same time the output of port increases, so the port does not re-enforce the output of the woofer, it REPLACES the output of the woofer at these frequencies. This, if done properly, can significantly reduce distortion and increase power handling at very low frequencies, a region that can be difficult for drivers. Vented systems can be up to 3 dB more efficient than a sealed box system that has the same bass cutoff frequency and size.

BANDPASS: These are compound systems in that they have at least two enclosures: one on the front and one on the rear of the driver. The enclosure on the front, which looks remarkably like a vented box (because it is), acts as a low pass filter, and, can couple the output of the woofer more efficiently to the outside. They have several useful advantages. For example, the front enclosure can be used as a very effective acoustic crossover, filtering out mechanical noises generated by the woofer, something no electronic crossover can do. For very low frequencies, such an acoustic crossover can be far less expensive and more easily designed than an equivalent electronic crossover. They are called "bandpass" because the combination of the rear enclosure and the driver form the high pass portion while the front enclosure forms the low pass section. Making the bandwidth of the system narrower, concentrates the available energy and raises the efficiency of the system.

COUPLED CAVITY: A variation of bandpass and vented systems, they are the results of a designers attempt to solve certain specific problems. They consist of two or more rear enclosures, each coupled to the next by a vent. Each enclosure/vent combination is another resonant system, and the combination is, essentially, a high order, multi-tuned resonant system. Generally, these systems have quite complex response and are difficult to design. No comprehensive theory on their operation exists like that for sealed, vented and bandpass systems.

What is the best material to make speaker boxes ?

An ideal speaker cabinet material would be very stiff, so that it would not tend to move with variations in box air pressure. It would also be very well damped, so that if it ever does deflect from air pressure, it will come back to the original position without resonating. It would also have a very high resonant frequency (supersonic), so that low frequency box air pressure would not cause it to resonate. An attractive material is preferred, and additional credit is given for a material which is easy to cut, glue, and finish. A great material would be cheap, too. Finally, it would be nice if the material were light, because we all have to move our speakers sometimes, and it's hard to appreciate good speakers with a sore back.
With all of those attributes, it would seem that no material is perfect. However, there are many materials that have enough of the above good attributes to make excellent speaker cabinets. Yet each has advantages and disadvantages.
In the list of good speaker box materials below, letters are used to indicate which attributes the material possesses. S = Stiff D = Damped H = High Resonance A = Attractive M = Malleable C = Cheap L = Light
MEDIUM DENSITY FIBERBOARD (MDF): SDMC This is the most practical material for quality speakers. It is resists interanal resonances better than plywood. Most lumber yards can special order it. It cuts very nicely and has a smooth surface. It takes veneer very well. However, bring a helper when you pick the stuff up. One sheet is very heavy. MDF is harder on tools than common wood, but easier than particle board. This is the material that many great speaker makers use. $45 for a 4'x8'x1" sheet. Density: 50 lbs/cu ft.
POLYCARBONATE (LEXAN): DMA clear or solid-color polycarbonate box can look strikingly good. However, this is not a cheap material. To locate it, look in the classified directory under PLASTICS. $400 for a 4'x8'x0.5" sheet. Density: 75 lbs/cu ft. Acrylic (Plexiglas) is cheaper than Polycarbonate, but weaker and poorer damped (not recommended).
PLYWOOD SHEETS SPACED AND FILLED WITH SAND OR LEAD SHOT: SDAMC If you have time on your hands and want a great impractical box, try this. Make a simple box out of common plywood. Then glue cleats on the outside of the box to space the outside plywood from the common plywood. Glue hardwood-veneered plywood to the cleats and pour sand or lead shot into the spaces between the cleats. It won't be light, but with the filler, it will be extremely well damped. In addition, if you use strong cleats and glue well, the box will be extremely stiff. Be sure to sterilize the sand in your oven before putting it in the box. This prevents the growth of mold and other extra sonic impedimentia.
ALUMINUM SHEETS FILLED WITH ALUMINUM HONEYCOMB (Aerolaminate): SDHL Airplanes use this material for flooring. The next time a plane crashes in your neighborhood, see if you can get the wreckage for your next speaker project. You can't get a better, lightweight material. If you're really ambitious, you can make your own sandwich out of high-quality plywood faces and a thick honeycomb core. You will probably need an epoxy to glue the honeycomb to the plywood. A home-brew sandwich is easier to cut and glue than Aerolaminate.
FORMED CONCRETE: SDHC Not a good choice for car speakers. There are tricks to working concrete, such as to cast braces, rebar, and steel-wire right into the mix. Also, some concrete is better damped than others. Remember to oil your concrete forms so that they can be removed. Most concrete speakers use an MDF front panel, but you can pour one if you use cardboard tubes or plywood rings to mold the concrete into the shape of a speaker cutout. Alternately, you can make a common veneered plywood speaker box and cast concrete inside it for stiffening. Any box can be improved by making the walls thicker, by bracing the walls, and by stiffening the walls. The stiffness of a material goes up as the cube of the thickness, so a slightly thicker material is much stiffer. A thicker panel will also have a higher resonant frequency because the stiffness goes up faster than the mass. Consider lining the inside of your speaker with ceramic tile, attached with fast setting mortar. You can get tile remnants cheaply. They are easy to apply and can be added as an afterthought to an imperfect box. However, be sure to attach all braces before tiling, because it is hard to attach anything to tile. Also consider bracing any weak parts of the box. For example, all joints will benefit from a wooden cleat. The back of the box will benefit from stiffeners where the speaker terminals are attached. Most importantly, brace the front panel, or make it out of a double thickness of material.