Excellent clarification from Al, with more specifics. No surprise there! Timbre77, you missed the important qualifying adjective of the amplifier scenario I described---"tube". As Al stated, while the power solid state amps create increases with dropping impedance, the opposite is generally true of tube amps, with the notable exception of the unique Music Reference RM-200, which actually behaves more like a ss amp in that regard.
As Al detailed, the different impedance taps on a tube amp allow the amp to provide similar power to all the taps---that's one task of tube amps output transformers. However, if your tube amp has 4, 8, and 16 ohm taps (typical in tube amps, though the RM-200 offers 2, 4, and 8 ohms), and you hook up an, say, 8 ohm speaker to the 4 ohm tap, the power available to the speaker will be less than it would be if connected to the 8 ohm tap. But, says Music References Roger Modjeski, a tube amp so employed will usually be producing not only less power, but also less distortion, and better sound. As Al mentioned, Roger calls this tactic "light loading". In addition to lower distortion, an additional benefit of using a lower impedance tap is that the amps output impedance will be lower---it will have a higher damping factor, and will interact less with the varying impedance characteristics of the speaker load, resulting in a more predictable frequency response.