The carrier, which alternates between positive and negative current, at any given time, makes one pair of diodes conduct, and reverse-biases the other pair. The conducting pair carry the signal from the left transformer secondary to the primary of the transformer at the right. If the left carrier terminal is positive, the top and bottom diodes conduct. If that terminal is negative, then the "side" diodes conduct, but create a polarity inversion between the transformers.
It is referred to as a "ring" modulator because the analog circuit of diodes used to implement this technique takes the shape of a ring. This circuit is similar to a bridge rectifier, except that instead of the diodes facing "left" or "right", they go "clockwise" or "anti-clockwise".
A particular elegance of the ring modulator is that it is bidirectional: the input and output can be swapped so the same circuit with the same carrier can be used either as a modulator or demodulator.
The ring modulator was invented by Frank A. Cowan in 1934 and patented in 1935 as an improvement on the invention of Clyde R. Keith at Bell Labs. The original application was in the field of analogue telephony, and it has since been applied to a wider range of uses.