What does an Amplifier do?
Basically, an amplifier is a device that boosts lower level audio signals to the levels needed to drive the speakers system. The signal boosting function of the amplifier is often metaphorically described as "A straight wire with gain". In other words, it should increase the quantity of the signal without effecting the quality of it in any way. In practice, it is impossible to do without adding distortion and noise from its own components, as well as boosting the noise from the input signal.
Distortion & Noise
Of the three main forms of distortion present in an amplifier, harmonic distortion is the one most often quoted. This is caused by the amplifier creating harmonics of the basic frequency it is handling. Interestingly, tube amplifiers tend to generate mainly even harmonics, as musical instruments do, whereas transistor amplifiers tend to generate mainly odd harmonics, which are far less pleasant to the ear.
Another form of distortion, crossover distrotion, is important to the end result, but manufacturers are usually reluctant to offer information about it in their products. Crossover distortion occurs in amplifiers that use different components to handle the positive and negative parts of the waveform (most do - generally referred to as class B amplifiers), and is caused during the changeover between the two.
Driving Speakers
I've already mentioned how damping factor effects the way the speaker performs. This is a function of the amplifier, so the choice of amplifier can partially control how well the speakers can do their job. Another very important interaction of the two is how well the amplifier can deliver it's power into the loading of the speaker. A real loudspeaker is a much more difficult load for an amplifier than the test loads used for measuring and testing. I have literally owned a highly-regarded amplifier that triggered its own protection circuits and turned itself off when faced with driving a certain model of quality speakers at low volume levels!
Power
Power is only a measure of quantity, not quality. The need is only for sufficient power to drive the speakers to the required volume without peak demands stressing the amplifier. For many listeners, the better option is often a higher quality, lower power amplifier rather than the other way round. Generally speaking, it is best to look at multichannel home theater amplifiers in the 100 watt or higher range.
What is the deciding factor for the amount of power needed is the combination of how efficiently the speakers can turn electrical power into sound, the size of room, and the volume level needed. Speaker efficiency varies enormously - some speakers need 100w of input power to achieve the same volume as others can with 10w. The size of the room also effects the power needed - not only do you tend to listen further away from the speakers in a larger room, but the volume of air within it that needs moving will be much larger.