Lookup "Gilbert Cell" and "Four quadrant multiplier".
More generally, look into "translinear circuits" which are more general, the multiplier being a special case. The key idea is the logarithmic relation between the base-emitter voltage and emitter current. Some number of transistors are connected with their base-emitter junctions in a loop. By Kirchoff's voltage law, the sums of voltages around this loop is zero, and by the logarithmic relation, you can say something useful about the product of emitter currents. With cleverness in making use of that fact, you can design a multiplier. Differential amplifiers are a relevant idea, just not the key idea.
and find the paper "Translinear Circuits" by Barrie Gilbert
and for a good read that covers much more but includes a nice little explanation of translinear circuits, look for the article "The Gears of Genius" in IEEE SOLID-STATE CIRCUITS SOCIETY NEWS, Fall 2007 issue. This article is followed by one describing a very fast multiplier circuit.