A simple BJT operated in the saturation region is a good start to understanding how a multiplier works: -
I'm not talking about the BJT operated in the normal region (where the collector current remains flat for large changes in VCE) - I'm talking sub 300mV between collector and emitter.
Look at the pretty colored lines between 100mV Vce and 200mV. Notice that they all have different slopes and these different slopes represent a different conductance (collector current divided by Vce). So, by varying the base current you can make a variable resistor. This is how linear BJT multipliers work. A multiplier is a modulator and you can modulate amplitude of (say an audio signal) by using a pot.
Unfortunately most op-amps can't be made to work in this way so they are not very useful for multiplication but I'm sure someone will point out one that is!
However, the simple BJT multiplier is not that great - you have to dc bias the AC signal so that it doesn't get distorted sub 50mV on the graph but it's a start and works pretty well for AM broadcast modulation: -
If you want full double side-band suppressed carrier modulation you need a four quadrant multiplier (double balanced mixer) and these can be fashioned by BJTs like this: -
One design I was looking at recently uses the SA602. It's a double balanced mixer and oscillator (combined) so it's pretty useful: -
When it comes to AM, op-amps aren't really what you should be looking for.