damping factor

The damping factor is the 'control' the amp has of the woofer movement.
When the amp has a signal that moves the woofer and then the other way, the damping factor is the control the amp has to "HOLD" on to the cone and not allow it to continue going in the first direction, instead of obeying the amp and moving in the opposite immediately that the new signal wants. (sloppy explanation but it will do...

So "IF" your speakers have a BIG, 12" or maybe even a 10" sloppy woofer, the higher damping factor amp will 'control' it better, and that big sloppy woofer will behave better, giving a more solid punch, with less overhang and less woolly sound.
Where if you own a set of speakers with a single or pair of 'tight' 8" or 6" woofers, then the damping factor will not matter nearly as much, and other factors will overshadow the damping factor.
Generally it is said ANY solid state amp has enough damping factor to control any modern speaker system woofers. Only if you had some old speakers with giant woofers..
And the place where damping factor may be a little too small for some speakers is in a tube amp. Although the combo of a tube amp and big efficient speakers is usually heavenly, even if the bass is a little sloppy.
So all in all, you do not have to worry in any way about that particular spec called Damping factor. It is used to sell stuff, particularly amps which 'just by chance' have a particularly BIG damping factor, but it is a nearly meaningless spec.

Actually, what happens is when the amp sends a signal to the driver it energizes the voice coil and causes the cone to move. When the signal stops, the woofer cone keeps moving by inertia (the amplifier cannot stop the moving mass). This causes the voice coil to move back through the magnet inducing an emf onto the voice coil. Ideally, the amplifier is to present a short to the speaker (via zero or very very low output impedance). If the amp has a high output impedance, it presents a load to the back emf, and by adding speaker cable capacitance into the mix, it causes resonance. It is this reflected, or oscillating, voltage that "looses control" over the driver. The higher the damping factor, the lower this effect. Does it mean better bass punch? All depends how you define "better".

Gs5556. Thanks, much better technical answer than my sloppy one.

But, particularly in the case of very efficient drivers, there is the issue of overdamping. Too much control over the driver can lead to a bass response that is less natural and not as deep as it otherwise could be. My Aleph 30 amp was great except that it was overdamped, causing an unsatisfactory bass response. I have since gotten an Atma-sphere amp with a lower damping factor that works much better with my speakers. As always, synergy with speakers is the key.

Good answers above. There is an additional factor that often needs to be considered, though. Damping factor and amplifier output impedance are inversely related. Damping factor, as usually defined based on an assumed 8 ohm speaker load, equals output impedance divided into 8 ohms. To the extent that the speaker's impedance varies with frequency, that output impedance will affect tonal balance, by interacting with the speaker's impedance vs. frequency variation.

Many electrostatic speakers, for instance, have high impedances at low frequencies, which descend to low values at high frequencies. A solid state amp, having a high damping factor and negligibly small output impedance, will produce LESS bass and MORE treble into that kind of speaker, relative to a tube amp. The tube amp, having a relatively low damping factor/high output impedance, will produce MORE bass and LESS treble into that kind of speaker.

Conversely, many dynamic (cone-type) speakers have low impedances in the bass and mid-bass regions, and higher impedances in the upper mid-range and treble. Driving that kind of load, a solid state amp will provide a stronger bass response than a tube amp, and the tube amp may sound excessively bright.

As was stated, synergy with the particular speaker is key.

Regards,
-- Al