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ANY box is a highpass filter. It allows the woofer to play louder at notes above the cutoff note or frequency, and filters out notes below that frequency at a constant rate.

A sealed box filters at the slowest rate. It has the least amount of time distortion. It is the least efficient, but it is often the smoothest response.

A ported box (of any type) is an enclosure that has a hole in it. The hole is like the hole in the top of a Coke bottle - when there's a sound, that hole makes one particular note louder than all the rest. This is the Helmholtz resonator principle.

Think of resonance like a bell. When you hit a bell, it makes one note louder than all the rest, and in the case of a bell, it's a LOT louder - and it resonates for a while. When you hit a piece of wood, it probably makes one note louder than the rest, but in most cases, it's not much louder, it's sort of a a dead sound, partly because it stops making that sound alomost immediately. (You may see where the time distortion might come from).

So when you port a box, you are saying, "let's make the air in the box vibrate at a certain note, instead of the speaker cone having to vibrate at that note, to get some output for free (almost) and extend the low notes that box/speaker combo can play.

If it's done right, that box does extend the low end without any funny peaks. If it's done wrong, you have funny, one-note bass, and if it's done really porrly, that on-note bass is so high in the bass that there's a lot of musical energy in notes below that note - and that will destroy your speaker. (Ported speakers lose all their power handling at a note 1/2 octave below the porting frequency, and below the ported frequency, the box has TWICE the rolloff rate of a sealed box - no such thing as a free lunch).

A bandpass box is a sealed or ported woofer box firing INTO a ported box.

The name comes from the fact that the external chamber acts like a lowpass filter too. A hipass and a lopass combine to make a bandpass filter.

Bandpass boxes are a way to make small speakers sound bigger than they are. This is why they are used by Bose in systems like the Acoustimass. (Having a ported box fire into a ported box is a design patented by Bose, and making such a box for commercial use, or telling someone how to make one for a woofer you sold them, are violations of US patent law. You can make one for your own use legally, though).

Since all the sound coming out the box has to fit through the port, port diameter is VERY important. If it's too small, you get port noise and a pretty hard volume ceiling, where the only thing that gets louder above a certain point is the port whistling.

When you make a port diameter larger, you have to make the box larger too, for the tuned note to stay the same.

So basically, in my experience, you can use these kind of boxes to generate sick amounts of bass - but they don't sound good from an SQ perspective, the port noise is really audible in hatchbacks, wagons, and SUVs (compared to being inside a trunk), they are large in size, and in their space needed you can put a better-sounding sealed speaker.

Other than that I have no opinion.