I'm not familiar with a 'headpiece' but I think you mean the bridle. This is used to control the horse's head and keeps the bit in the horse's mouth.
Horses did not live whith Knights.
they fighted on horses
One of the original limiting factors on the amount of armor a knight could wear was the weight his mount could carry. As warmbloods (draft horses crossed with Arabian-descent horses) were developed, knights got larger horses that could carry more weight and still be useful in battle, allowing for the knight to wear more armor.
tall and strong ones like quarter horses. Depends on the knights.
knights rode horses like any other person would ride a horse
knights wear armor such as mail that protects their upper half
Horses
Friesian horses were popular war horses, and mounts of knights. For the Howrse game for Archimedes the answer is "for carrying knights".
A "bridle path".Makes a channel in a thick mane for the headpiece of the bridle to sit in.
The squire. The squire would carry the knights weapons or armor, hold any prisoner the knight might take, protecting the knight, and take care of the knights horses.
It is called a bridal patch or bridal path for the headpiece of the bridal to sit when it is on.