What is it?
A type of loudspeaker enclosure. The most common form is a rectangular box, airtight, except for the two big holes in the front. Set into one hole is the woofer, and set into the other is a tube or "port". When the driver responds to an electrical signal, it flaps about merrily, pushing at the air on the inside and on the outside of the box. As the air inside the box gets alternately stretched then compressed, it sets up a vibration in the port, which bosts the driver's output at some frequencies, and diminishes it at others.
Part of it is to do something useful with the back wave - sound that is radiated from the back of the speaker. The box usually contains some stuffing to soak up the high frequencies, and the volume of the box and size of the port are set so that the noise made by the port slightly extends the bass response of the driver, compared to a non-ported enclosure. The port works as a Helmholtz resonator; it is exactly the same effect as the one you get when you blow over the top of a bottle.
Why aren't all speakers ported then?
Well, most are, I think. It's the easiest way to get impressive sounding bass out of a driver. But a simple sealed box speaker is better in some regards - mainly in subtle ways, which are harder to cash in on than the obvious dollar value of speakers with PHAT BASS.
Compared to sealed speakers, bass reflex types are a complex resonating system. Starting and stopping those resonances takes time, which makes their transient response relatively poor. Also, the bass output is less tuneful, as the port only resonates at one note. All the sounds around this frequency get a bit blended together, rather than being played as distinct tones - this is quite benign in good designs, and rather horrible in others. Finally, below the port frequency, the speaker runs into real trouble, and the bass drops off very sharply, whereas a sealed speaker will roll off quite slowly.
What makes some better than others?
There is obvious stuff, like the quality of the driver and the enclosure. Then there is sheer scale. As a general rule, bigger reflex boxes sound better than small ones, as their ports are tuned lower, and the region where transient response is degraded occurs at lower frequencies. This is important because our hearing is most acute in the midrange, from approx 300-3,000Hz. The further from this range a sound is, the less well we hear it, so distortion way down low is less harmful than distortion at a frequency near to the midrange.