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.
