Gain is a control on an amplifier that adjusts the level or strength of the signal sent through the preamp stage. It can be thought of as the input volume to the preamp stage, although its more of a tone control than a volume control. The gain setting determines how hard youre driving the preamp section of your amp, and setting the gain control sets the level of distortion in your tone, regardless of how loud the volume is. Gain is expressed in decibels (dBs), and it multiplies the input signal.
In summary, gain is a control that adjusts the level of the signal sent through the preamp stage of an amplifier, and it determines the amount of distortion in your tone. It is not the same as volume, which controls the final output signal somewhere along the way.