You have three basic types of amplifier. Where you draw the line is a little arbitrary but this is the idea. You have:
Pre amp
IPA Intermediate Power Amplifier
PA Power Amplifier
With audio, IPAs are not necessary. Neither is power transfer. With modern electronics, for audio we are concerned only with Voltage transfer. Hence, matching impedance is totally unnecessary unless you are making a telephone system with miles of wire.
So, given that we are talking guitar amps, the Pre amp purpose is to raise the voltage of the source (in this case a guitar pickup) to a leval useful to the Power Amp. We have then raised the Voltage but have not yet given the ability to provide current at the voltage to develop useful power. It is easier and cheaper to add equalization and effects at low power so manufactures alway put equalization and effects in the preamp. It could be added in the power amp but would result in unecessary expense. Now that we have adequate voltage and all our equalizing and effects, we need power. So, we feed the output of the Pre Amp to a Power Amp. The Power Amp usually does not increase voltage but at unity voltage gain (ie no increase in voltage) adds the ability to supply current sufficient to drive a speaker (Typically 4 or 8 Ohm) If you try to drive a speaker with the output of a Pre Amp, the Pre Amp will be unable to supply the current to maintain the Voltage and hence the Voltage will drop and little or no sound will come from the speaker. The final Pre Amp stage may even overheat and destroy itself. This is why we have both Pre and Power Amps. Intermediate Power Amps are typically used in Radio where not only are we transfering voltage but indeed power.