Before we define some amplification terms, a little more on heat. This is not only because larger amps take up more room. Once this is accomplished, the signal is delivered to the speaker terminals at the other end of the amp. Speaker wires connected to the terminals then carry the amplified signal to the speakers. Amplifier output is typically measured in watts per channel, with a normal stereo system having two channels, left and right.
More watts means more power output to the speakers, which in turn means louder volume.
Be aware, though, that in the aftermarket, and even at the OEM level, manufacturers will frequently massage the numbers to make their amplifiers appear to put out more power than they do. As ludicrous as this sounds, it's a very common practice in the industry. Not in any world you or I populate. This is a snapshot of the amp in one particular moment of performance. Any decent audio amplifier will produce two, three, even four times its RMS power for a split second.