Everyone concerned with buying audio and video gear has heard the term Impedance and most people know that the impedance of the amp and speaker must [or should] be matched. Why? Because the amps output stage, the speaker and wire connecting the two constitute a transmission line and transmission line theory states that a wave propagated from the source [amp] must be properly terminated at the destination [speaker] in order to work properly. Matching the impedance of the two accomplishes that. Otherwise anomalies, such as reflections, phase errors and frequency specific cancellations can and do occur, not to mention the fact that the efficient transfer of power from the amp to the speaker is reduced.

So what exactly is impedance? Simply stated it is the opposition to the flow of alternating current [AC]  in an electrical system and is comprised of 3 parts. Two parts are termed "Reactance" [X} for the AC elements and one is termed Resistance [R] for the DC element. So restated Impedance is the complex Reactance and Resistance to the flow of Alternating Current, which of course is what analog audio is. Impedance is symbolized with the Greek omega sign [the same as for DC resistance] and is referred to in Ohms as well. I said complex earlier because the two elements of Reactance [X] are Inductance [L] and Capacitance [C] and they are in opposition to each other electrically.

An Inductor will appear to Direct Current as a short circuit, presenting no resistance at all, but as the waveform rises in frequency from zero to infinity, an inductor looks increasingly like an open circuit, or said another way, presents higher and higher opposition to current flow. Your speaker's voice coil in the driver is highly inductive, as is an amplifier's output transformer, if used.

A Capacitor is the opposite of an Inductor, it is an open circuit to DC and looks more and more like a short circuit as frequency rises. It is these very characteristics that make capacitors and inductors ideally suited to filtering applications. Both capacitors and inductors are used as filters in power supplies and equalization/filter networks. - another subject. In the speaker/amp combo both elements are present: Inductive Reactance [XL], Capacitive Reactance [XC] and in addition, resistance [R] that form the complex relationship called Impedance [Z].

Further, and here is where it get a little gnarly, in a purely Resistive circuit current and voltage are said to be "in-phase" meaning as voltage varies, so does current in direct proportion and at the same moment in time. The in-phase statement is not true for Inductive and Capacitive [reactive] circuits. In these circuits voltage and current are always out of phase. In the purely Inductive circuit voltage leads current by 90 electrical degrees, and in a purely capacitive circuit voltage lags current by 90 degrees. So these two components [L and C] are electrically 180 degrees apart, one being +90 degrees and the other -90 degrees. This is a graph illustrating these relationships.

When the capacitance and inductance are exactly equal [doesn't happen very often in the real world] the Reactive elements produce a sum that makes the circuit appear to become purely Resistive, implying that voltage and current must be in phase - which is true.  So now you can see why it's a complex equation for calculating Impedance. It gets much deeper, such as 'real' and 'imaginary' currents [ j ], power factor [pf], real and imaginary power, angular frequency, etc.

But interestingly, speaker impedance is not fixed in the real world, it varies with frequency! So your speaker might be 4 ohms in the mid-range but could be 8 ohms above 10Khz  and only 2 0r 3 ohms below 50Hz. So matching impedance is really a job of matching the average impedance across the spectrum. When a manufacturer states an impedance for a speaker they invariably state it as "nominal". The same is true of amplifier impedance. When I was still in the pro studio business there was a time when we kept blowing up amps and we couldn't figure out why. After a lot of study [and smoke] we found that the speaker-amp combo was dropping to near zero impedance at very low frequencies - basically a short circuit, frying the amp on loud, low passages. Thankfully that was the first and only time I have witnessed such a thing but it illustrates how Impedance varies with frequency.

The above basic discussion addresses electrical impedance which is quite different  from acoustical impedance, which deals with sound through air, e.g. waves, sound pressure and particle velocity,  etc.

Hope this useful information. Feel free to contact me with questions.