A brass ring is a small grabbable ring that a dispenser presents to a carousel rider during the course of a ride. Usually there are a large number of iron rings and one brass one, or just a few. It takes some dexterity to grab a ring from the dispenser as the carousel rotates. The iron rings can be tossed at a target as an amusement. Typically, getting the brass ring gets the rider some sort of prize when presented to the operator. The prize often is a free repeat ride.
The figurative phrase to grab the brass ring is derived from this device.