What does a horses headpiece do?

A "horse's headpiece" is a broad term that could refer to several different things. To give you the best answer, I need more context. Could you please clarify what you mean by "headpiece"?

For example, are you talking about:

* A bridle: This is the main piece of equipment that goes on a horse's head and is used to control the horse. It includes a bit, reins, and a headstall.

* A halter: This is a simple headpiece that goes over the horse's nose and is used for leading and tying.

* A noseband: This is a part of a bridle that goes around the horse's nose and is used to help control the horse's mouth and head.

* A browband: This is a decorative band that goes across the horse's forehead.

* A fly mask: This is a headpiece used to protect the horse's face from insects.

* A headcollar: This is a type of halter used for specific purposes like leading horses in shows.

Once I know which type of headpiece you're asking about, I can give you a more accurate explanation of its purpose.