What’s the relationship between gain (dB) and power (watts)?

Awesome question and one I’ve been contemplating myself.

and awesome answer MC, thanks to the op and the og

None. :) 

Amps tend to, by convention but not requirement, have around 28 dB of gain, regardless of their power rating. :)  This is about 20x input voltage.

That is, put in 0.1V peak to peak input signal and you should get 2V peak to peak on the output.

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.  So, if your rails are +- 20V, that's your peak output.  With 20x gain (around 28 dB) this means your maximum input voltage is 1v peak to peak before clipping.

Thanks everyone.  I like the gas pedal vs engine analogy.  But does a higher initial gain ultimately mean anything for the output of the amplifier (like higher possible SPL in dB for a given speaker)?

Yep, but actually your max usable output is about 0.71 of your rail voltages.

Gain is the amount of gain the amplier has usually between 20 to 30 db and power is the amount of watts it will put out into an impedance in ohms. The higher the gain the more loud it will be at a certain volume setting on your preamp or amp but with the higher gain more noise will be introduced but if you have efficient speakers  you will always want less gain to lower the noise and use more of your volume control. With less efficient speakers you may need the extra gain because you will run out of volume control too quickly so it is for system matching. Power is a relative thing that also depends on how much current you have in the circuit, tube amps are high voltage low current, and solid state amps are low voltage high current, so different types of speakers will like one or the other.