Question: Is there a general 2-input/1-output circuit that multiplies the voltage of one wire by the voltage of the other wire?

For example, Voltage of A = 3, Voltage of B = 5, output voltage is 15.

Answer: 
There are several possibilities, mainly depending on required accuracy and speed (signal frequency)

* dual gate MOSFETs (e.g. BF961)
* mixer ICs e.g. NE612
* precision analog multiplier ICs e.g. AD633, MPY534
* log-add-antilog OpAmp circuits (i.e. calculate logarithms of both inputs signals, add them and calculate anti-log (exponential function) of sum). The log and antilog functions can be accomplished by OpAmp circuits that exploit the exponential I-V characteristic of semiconductor diodes.

The first two solutions are good for RF and when scaling is not important (e.g. RF mixing).
The latter two are good when scale is important (e.g. in analog computer). 

For special cases when one of the input signals takes only values 0, 1 or 1, -1 you can use analog switches.
For the simplest approach, you can use a **variable-gain** or v**oltage-controlled amplifier**. 

Furthur as mentioned by Jack B in above comment, you can select an IC for [analog multiplier](https://en.wikipedia.org/wiki/Analog_multiplier) depending on the use in power electronics or signal processing electronics.