Damping Factor

It is literally just a number,  it is eight divided by the output impedance of the amp.  The number 8 was chosen as the typical speaker impedance.  If your amp has an output impedance of one ohm, then its damping factor is eight.  This number is suppose to signify how your amp will react with the impedance characteristics of your speakers.  The higher the damping factor (i.e., the lower the output impedance of the amp) the lower the amount of effect that the output impedance will have on the frequency response of the speaker.  Sonically, the higher the damping factor the more it would seem that the amp has control of the bass response--tighter bass, more punchy bass.

The real issue is how much damping factor is sufficient.  Probably eight is plenty enough for most speakers.  Low impedance speaker might benefit from a higher damping factor.  Beyond what is sufficient, damping factor is largely meaningless.  Tube amps always have a much lower damping factor than solid state amps. 

Solid state devices inherently have a low output impedance.  The output of tubes is low current and high voltage (also meaning high impedance) and an output transformer converts that high voltage to higher current and lower impedance.  But, there are practical and sonic reasons for limiting the extent to which the transformer lowers output impedance.  That is why many tube amps have output impedances higher than one ohm, and therefore, damping factors less than eight.  That has not stopped such amps from being among the best sounding.   

One of the problems with determining how important is a high damping factor is the "all other things being equal" problem.  You never can achieve all other things being equal, so it is hard to say if any difference is attributable to damping factor.  Also, there are some systems where low damping factor does indeed alter the sound, but, it can be for the better.

How high damping factor is achieved matters a lot to me.  Throw in a lot of negative feedback and you can get spectacularly low source impedance (high damping factor) but those designs tend to sound dead, or "sterile" to my ears.  I recall when Halcro amps came on the market and audio magazines went nuts raving about them.  They measured spectacularly--ultra low distortion, utlra high damping factor, etc. and I did not like them on any of the speakers I heard them playing.  A friend of mine said that he had tears in his eyes when he first heard them play because of the nostalgia it provoked--it reminded his so much of the first time he heard a Phase Linear 700 amp (i.e., he thought it sounded awful too).

If deep and tight bass were the be-all and end-all of amplifier performance, then only solid state amps would be in the market (cheap ones can deliver this).  But, there is so much more to performance, even in the bass range.  I have no idea if tube amps that are not as tight sounding are coloring the sound and  adding in distortion, I only know that I like the upper bass performance of many lower-powered tube amps -- bass has a better sense of subtle and varying "tone," it has less of a dry quality and "mechanical" sameness.  I hear the same sort of dry and toneless bass with many high-powered tube amps, so it is not necessarily a solid state vs. tube issue.  The amps I tend to like probably have a low damping factor, but whether this is the "cause" of the sound I like or whether it is just something that attends the basic designs that I like, I don't know and don't really care.

In short, I don't think damping factor means much at all, and it certainly isn't a high priority consideration, beyond some minimum that affects basic compatibility with a given pair of speakers.  I would not worry about damping factor at all if I had genuine 16 ohm speakers.  With an 8 ohm speaker, I might be concerned with an amp with a damping factor below 8 (but I would still audition because there is a good chance the combination will still work out).  With 4 ohm speakers, i might be a little concerned with something below 15.  I've heard SET amps with 4 ohm speaker that sounded terrific, and many of these don't have have damping factors above 4.