What do you actually mean by DC voltage gain and AC voltage gain?
"Gain" means how much bigger is the voltage (or current, or power) at the output than at the input.

Gain is important because it allows us to take a small signal and amplify it to make a larger signal. You might want this because, for example, a larger signal will be easier to detect with a analog-to-digital converter, or it might be able to be detected at a farther distance when broadcast as a radio signal.