Some very interesting thoughts on this thread:
I will keep this simple because there is no need to complicate this unless you are building your own system.
Matching Power
OK, the Amp can be seen like a car engine in that it supplies power, and your speaker like the clutch brakes and tyres delivering the power to the road. If you put a bigger engine in it will work, but your clutch will fail much quicker if you use all the power and the tyres and brakes will fade and you will be replacing parts much sooner than you would like!
The same is true for the amp and speaker, drive the speaker past its limits and you will be replacing the speaker a lot sooner, so rule of thumb is make sure your speakers can take all that the amp can give plus a bit more.
Now if you put a smaller engine in the car again it will work but perfomance will be sluggish.
Using a speaker that needs a lot more power to drive it than is being supplied and it will not reach its optimal range, but it will work.
Matching Impedance
If you are running multiple speakers make sure that they are of the same impeadence (ohm's), if not your sound will not be equally divided, (ie one will be louder than the other), as the power will be split based on the resistance that each speaker gives, and if they are different then the output of each speaker will be different.
If the speakers are the same ohms ie 8ohms and 8ohms = 4ohms then the power, (watts) will be divided equally, so if your Amp says 500watts at 4ohms your lowest speaker should be rated greater than 250watts.
Volume Controls
How much output, (watts) being delivered to the speaker, (not to be confused by being generated by the amp) will depend on where the volume control is set, and there is no set relation to the number on the dial, (so just because it goes up to 11 does not mean that it is louder than one that goes up to 10, remember Spinal Tap).
So, bottom line is if your Amp is more powerfull than your speaker keep the volume down, and your speaker may just last out, (although I would not recomend this as a long term arrangement).
I am sure that this will generate a bit of further discussion as in this case fiction is greater than fact!