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)?

@millercarbon - thank you.  I like this:

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.

@timlub - thank you. This is helpful and simple:

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.

@erik_squires - now this seems relevant to me:

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.

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:

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.

@millercarbon

Gain is a multiplier, power is a ceiling.

a ceiling of what, though?  Current?  Voltage?  

@rixthetrick - thank you sir!  Love your posts and will refer to them again as needed.

Amplification can be somewhat described as if you had a small electrical signal controlling a valve on a larger electrical signal, as the smaller signal swings from positive to negative, in higher and lower amplitude from zero, it opens and closes the valve proportionally, positive and negative. 

@erik_squires - Thank you. That’s helpful.

Well, think about how much an amp can do. Amps can't have infinite power, or output voltage, right? They are limited devices. Every one of them. What limits them? Among many things, the voltage available to the output circuits. That is the absolute limit of output.

Your gain is 20x (28 dB) for instance. OK, but your maximum voltage out is 10 Volts peak to peak.

1 volt pk-pk in x 20 = 20 Volts pk-pk.

With this amp, 20 volts out is not possible.   You need a bigger amp. What you'll get instead is a clipped signal that goes up to 10 and hangs there until the input voltage goes back down.

@pragmasi - thanks for that extra holistic perspective.  Interesting to think about noise and dynamic range that way, but I have no idea if you're correct. 

Is it true then that the output stage of the amp 'transforms' the amplified signal from the gain stage (in voltage) to current (in amps)?

The bigger the power supply that's available will allow for a bigger/wider peak voltage swing to be transformed into current to the speakers before it clips? Thus, power equals headroom?  Lot of gain + minimal headroom = a limited amplifier. 

@pragmasi - I appreciate the layperson approach. I don't think I'd really be able to take in a fully complex explanation right now. Metaphor and analogy much more my speed.