An ornament, usually circular, through which are drawn the two ends of a woman's or man's scarf, the wearer then sliding it up to near the neck. Sometimes there are two adjacent rings ornamentally joined. Also called a 'scarf ring'.
From: An Illustrated Dictionary of Jewelry, autor: Harold Newman, publishers: Thames and Hudson
We received an email telling us that there is also another explanation for Belcher ring; it is the following:
Although the dates are Victorian, this is a uniquely American style, named after American designer Thomas Belcher. The defining feature is how the stone is secured into the ring -- using prongs or claws that are cut into the shank of the ring, so that the stone does not extend above the circumference of the shank. A secondary characteristic is that most Belcher-style mountings have a round dip between the prongs at their base. The closest European counterpart is the flush-set “gypsy” mounting. It is a sturdy style, and because the stone sets low into the ring, it is less likely to catch on things