Curb chains are part of a couple of the bits for a horse's bridle.
The bit is the metal part that goes in the horse's mouth, to which the reins are attached. In a sense, it provides the rider's brakes and the steering for the animal.
There are different sorts of bits, such as snaffle, pelham and curb.
The curb and the pelham both use a curb chain. The curb bit consists of a bar through the horse's mouth, which has a bit of a kink in the middle. Outside it has straight metal cheekpieces, and a metal hook on each side to which the curb chain is fastened. This passes around the back of the horse's chin. The pelham is similar, but without the kink in the straight mouthpiece.
You can see some examples here.
https://www.thehorsebitshop.co.uk/section.php?xSec=18&xPage=1