Sometimes this stuff just gets too complicated. Think of it in terms of batteries and light bulbs, and imagine for the moment that we have a simple 2-volt battery has all the power we want available, but there is a 1-ampere fuse in the circuit to prevent us from burning up the wiring we're using. For practical purposes, an amplifier and its speakers behave like the battery and light bulbs since in speaker systems impedance is roughly equivalent to resistance.
A 2-volt battery will deliver 1 watt of power into a 4-ohm light bulb (Power=Voltage 2 divided by Resistance, 1=22/4, 1=4/4, so a 2-volt 1-watt light bulb will have 4 ohms of resistance). Note that when delivering that one watt, a current of 1/2 ampere will flow (Current=Voltage divided by resistance, 0.5=2/4=1/2).
If we put two 4-ohm light bulbs in parallel, each will receive 1-watt and ½ ampere will flow through each. That would give us a total of 2 watts with 1 ampere flowing--the maximum our fuse will allow. Note also that the resistance of two 4-ohm light bulbs in parallel is 2 ohms.
In terms of doing useful work (producing light or producing sound), a 2-volt source is a 2-volt source whether it's coming from a battery or an audio amplifier. Here's how much power that 2-volt source can deliver:
Remember we said there's a 1-ampere fuse in our system. Based on the above, with two 4-ohm speakers in parallel, the amplifier is delivering that one ampere or two watts total to the two speakers. We could say it's a 2-watt amplifier in this case.
Now suppose we only have the one speaker (4-ohms with ½ ampere flowing at 2 volts). Now we can raise the voltage (turn up the volume) to 4 volts to get 1 ampere flowing. Now the amplifier will be delivering 4 watts with the same 1 ampere flowing and we won't blow the fuse.
If we connect an 8-ohm speaker instead of the 4-ohm speaker, we suddenly need 8 volts to get to 1 ampere of current, but now the power is 8 watts. Keep in mind that if we put our 4-ohm speaker across that 8 volts, the power is now 16 watts--for a split second before the fuse blows. If the amplifier could handle more current (a higher-capacity fuse) it could deliver more power, at some point enough to destroy the speaker instead of the fuse.
Modern amplifiers may not have a fuse but instead have some kind of equivalent protection mechanism, though some can be damaged through excessive current in their load. They are rated in watts, usually the number of watts they can deliver into some load, usually 4 or 2 ohms. So our sample amplifier above might be rated at 2-watts into 4 ohms, since the fuse will blow with anything over than 1 ampere flowing. With the same amplifier capable of delivering 4 volts into 8 ohms, it could be called an 8-watt amplifier. In effect, amplifiers are both current- and voltage-limited. There is maximum current and voltage they can deliver without damage or shutdown. Since voltage multiplied by current is power (watts) that's a way to rate the amplifier.
Speakers have power ratings, too, but they are not matched to the amplifier except that the power rating of the amplifier should not be greater than the total power rating of the speakers connected to it.
Most amplifiers are capable of delivering more power into a 4-ohm speaker than an 8-ohm speaker for the reasons describe above, but that doesn't mean 4-ohm speakers are necessarily better any more than 120-volt light bulbs like we use in the US are better than 240-volt light bulbs used in Europe. It's also not true at all that a speaker with a lower impedance requires a more powerful amplifier. A typical amplifier will usually deliver more power to a lower impedance speaker, all other things being equal.
To address your headphone question, you can think of an amplifier as a voltage source rather than a power source. A lower impedance placed across a fixed voltage will absorb more power, so lower-impedance headphones will be louder when directly connected to a particular amplifier. That said, however, headphone distribution systems vary widely and there is often a current-limiting resistor in series with a typical headphone jack. The reason for this is to protect you: A pair of low-impedance headphones connected to a powerful amplifier could easily be damaged. More importantly, your hearing could be seriously damaged. A high value current-limiting resistor will make high impedance earphones seem louder than low impedance phones and vice-versa. In practice, you don't want headphones to ever receive more than a small fraction of a watt from an amplifier because they will be dangerously loud.
Hope this helps.