Curbs are used on leverage bits, which are used by most western riders and at higher levels in dressage and hunting. Some driving bits also have curb chains.
A true leverage bit cannot function correctly without a curb. The curb chain is often seen as making the bit "stronger" but it also serves the purpose of stabilizing the bit in the mouth and preventing the bit from rotating excessively, which can injure the roof of the horse's mouth. The shanks should never rotate more than 45 degrees. Mechanical hackamores also have a curb chain.
This is a western curb bit with a leather strap. Source Montanabw via Wikimedia Commons.