Amplifiers are a vital component - while the loudspeaker reproduces the sound with a varying degree of accuracy, a speaker can only be as accurate as the signal sent to it. While there are an endless supply of meaningless buzz-words attached to almost all amplifiers, there are meaningful numbers as well.
Perhaps the most commonly flaunted number is "watts RMS". This is, very simply, a measure of power - in other words, what would happen if the signal were rectified to DC from an AC sine wave, and the wattage (volts multiplied by ampres) were measured. While there are several other forms of measurement also employed today, such as "music power" or "PMPO", these are almost always meaningless, and should be treated as marketing baloney.
An amplifier's RMS output is usually stated at a given impeadance - for example, 120w into 8 ohms, or 200w into 4 ohms. This is basic electronic physics - current is equal to the voltage divided by impeadance (ohms), and if a given voltage is applied to half the impeadance, the resulting current - and hence, wattage - will double. This only occurs in the best of amplifiers, though: with the exception of some massive Nelson Pass designs, inefficiency and internal resistance within the amplifier - if not simply the internal power supply - will limit output to a bit less than double.
Some cheaply-made amplifiers are only capable of driving an 8-ohm load. While it's usually safe to use them with a 6-ohm load (common on some home-theater speakers), a 4-ohm load will cause too much current to flow through the output transistors on the amplifer - which will, in turn, cause it to overheat and die.
As much as it is flaunted, raw power is not a measure of quality. Much like speaker drivers, amplifiers also have a measurement of distortion - but they must also include residual noise from the power supply and crosstalk. This is abbreviated to THD+N, and is the benchmark for audio quality.
However, there are several tricks in measurement to look out for. First, almost all but the highest-quality amplifiers distort more at higher frequencies; as such, the THD+N is measured at a very low frequency, and this may give misleading results - especially on some cheaply-made class-D amplifiers. Secondly, distortion will often increase as the power output increases - make sure that the THD+N rating is at no less than 1/10th the rated maximum output power, or roughly half maximum volume.