Damping is a measure of the rate at which vibration energy is dissipated in a system. If you're thinking about AC current, then damping is analogous to resistance in a DC circuit. It's often expressed as a 'Quality factor' or 'Q-value', with high Q values denoting low loss. A material that has a Q=100 dissipates 1% of the energy of the system (ultimately as heat) per cycle of vibration. Since this works like compound interest, after ~72 cycles the vibration energy would be 1/2 of what it was at the start, or 3 dB lower. Note that if it's vibrating at 100 Hz that takes .72 seconds, while at 1000 Hz it's .072 seconds, ten times faster. Tapping on a piece of high Q wood, such as typical Brazilian rosewood (Q~150) give a longer 'ring' than a lower Q (higher damping) wood such as maple (Q~70).
Another noticeable thing about high-Q systems is that they give a more secure sense of pitch. Low Q systems are harder to drive to high amplitude, even at their resonant pitches, but, by the same token, they're not much harder to drive off resonance than on it. The response is spread out over a wider frequency range, so you don't get the strong impression of pitch with a low Q material. They make xylophone and marimba bars out of rosewood rather than, say, oak (Q~70-100) to get the better pitch. Aluminum has low damping (I've measured Q~ 10,000 or better): it makes good tuning forks and 'vibraharps'. Low damping systems make good time bases: quartz crystals run up in the million range.
It seems to me that when you're talking about a guitar the damping of the wood is only a part of the story. Basically, I thing it sets a limit: you could not make the instrument have a higher Q value than the material it's made out of. Sadly, it seems to be all too easy to end up with higher loss, lower Q. Using the best wood is not a guarantee; the maker still has to avoid messing it up.
