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

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.

For me the easiest analogy would be....  With higher gain,  it takes less of a turn of the volume knob before you hit the amplifiers full output. 
With the lower gain setting,  you need to twist it a bit further.  

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)?

Yeah I was afraid of that, as it is a horrible analogy. So awful in fact I am gonna leave it alone and just explain what gain actually is, in the hopes the truth will replace all the false ideas generated by that awful example.

Gain is nothing more than another word for multiplier. Gain is the amount by which the incoming signal is multiplied. Because gain is a multiplier it is expressed in dB. So in your case, say you have selected 23dB gain. Whatever voltage level is input, is increased by 23dB.

Remember, dB is a log function. Remember, every 3dB is twice the power. Every 10 dB is ten times the power. So your 23dB equates to 200 times. (The math in this case is straightforward. But if you need help, ask!) Therefore, say your input voltage is .1mV. With 23dB gain your output voltage will be 20mV.

Flip the switch to increase the gain to 29 and since 29 is 6dB more than 23, that is two 3dB doublings, the output will be 4X the previous example or 80mV.

That is gain. It is like a calculator that no matter what number you enter all it does is multiply by 8. Or 16. Or whatever.

However, what if the calculator has a display with only 2 digits, and you input 100? Well clearly the gain does not change but the output cannot go that high, you are not going to magically get three digits no matter what the gain.

That is the difference between power and gain. Gain is a multiplier. Power is a ceiling. You cannot get any more power from your amp no matter what gain you set it on.