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A diode mixer exploits the fact that the voltage across a diode is the log of the current thru it, to a good enough approximation for many uses.  Now note that multiplication can be performed by taking the log of two values, adding the logs, then doing the un-log (take the exponential) of the result.  Another way to say this is that multiplication in linear space is addition is log space.

A diode multiplier starts with two current signals as input.  These are run thru diodes to get voltages that are the log of the input signals.  These are added, and the result applied as voltage to another diode.  The current thru that diode is then, in theory, proportional to the product of the two original input currents.

For signal mixers, you can often be more sloppy because the extra harmonics generated by a more sloppy process can be filtered out later.  Sometimes you are only looking for the resulting amplitude, and know the frequency quite well.  In that case, a tuned filter on the output can compensate for a lot of non-ideal characteristics of the multiplier.