They might be made of wool or medieval fustian and were lined with softer materials such as satin, silk or linen as a shirt would not be worn underneath. Sometimes the material would be "cut full of holes" for ventilation. Red was a popular choice for this garment.
Leather was used to reinforce the garment and provide attachgment points for defences to be attached. These defences might include mail voiders which go under plate armour to act protect exposed areas. In the event that the voiders are made separate and removable, they can be easily removed and then cleaned (as can be the doublet) which resulting in large amounts of corrosion of the mail.
The arming doublet would be padded in areas where blows were expected to fall, particularly the shoulders and where Charles du Blois expected shoulders to be padded to a thickness of "four fingers". Reconstructions have shown this to be a sensible precaution.
When fitting plate defences it would be a wise precaution to contruct this garment first, fit voiders etc, before ordering the rest of one's harness.