This is a great question. The not very satisfying answer is that it depends.

There are a wide variety of fabric armours and armour-related garments that can be described as quilted (that is, having an inner and and outer layer sewn directly to each other in the interior of pieces of fabric, rather than just at the edges). Indeed, most fabric armours seem to have been quilted in some way in medieval Europe. These are called variously gambesons, aketons, jupons, pourpoints, jacks, doublets of fence and arming doublets, names to which we can generally attach some general form and purpose but which we can't pin down exactly. This covers garments warn through hundreds of years, of many different shapes and fits and styles and compositions and worn for different purposes. So broadly speaking, these garments were constructed in a wide variety of ways and out of a wide variety materials.

Firstly, these garments could be either stuffed or layered. In either case the outer layers would be joined with quilting of some form in most cases. In some cases where thinness was desired, there might be only two layers, in which case the quilting would only provide extra sturdiness so that the fabric wouldn't warp (more on that later).

So, first, the diversity of -purposes- of garments that we can describe as 'quilted'. Some quilted garments weren't even related to armour at all - there is some discussion as to which surviving garments, like the Charles de Blois pourpoint - are actually military in any way, and at times wearing quilted garments for non-military purposes appears to have been the fashion. The generic term 'pourpoint', which seems to have referred to a variety of quilted garments for a variety of different purposes, seems to have referred to these garments as well. Secondly, they could be worn under armour. The origins of using a dedicated garment under armour aren't clear, but it was done by the 13th century or so and continued, in different forms, for as long as armour was worn. Early under-armour garments might be called an aketon or perhaps sometimes a gambeson, eventually they were called arming doublets. Thirdly, quilted armours could be worn -over- other types of armour like mail or plate armour. In the 14th century, this was the jupon. Later 15th century jacks were also worn over mail armour as well, though the fashion of wearing padding over plate seems to have mostly stopped. Finally, quilted garments could be independent armours. This describes most 15th century jacks, probably mostly describes 13th and 14th century gambesons (the exact distinction between gambeson and aketon is fuzzy) and also describes 15th century 'doublets of fence.'

Now, why do these purposes matter? Because they effect the form of the garment. Getting back to your question, the exact material used to stuff, or whether the garment was stuffed or layered, how many layers/how much stuffing would depend upon the purpose. Civilian garments would be padded out as much as was fashionable, while under-armour garments wouldn't be too thickly padded so that you didn't require excessive 'sizing up' of the armour worn, and to allow free movement within the armour. Indeed, after plate armour was more fully developed in the 15th century, arming doublets seem to have been rather thin, perhaps as few as two layers without stuffing, with the padding mostly providing stiffness so that the garment wouldn't pull apart under the weight of the armour that was tied to it. Over-armour quilted garments could be thicker and looser, while independent fabric armours were the thickest of all. Both 15th century jacks (recorded in the ordinances of the King of France) and 15th century 'doublets of fence' (recorded in the Howard accounts) seem to have been over two dozen layers of cloth.

The purpose would also affect the 'hardness' of the garment - independent armours might be quite stiff (especially if they included leather as an outer layer, as they sometimes did), while under-armours and civilian garments would need to be soft for comfort.

As to stuffing types, I have seen references to tow and raw cotton (which seems to have given the aketon its name). The example I know of straw being used is a helmet liner for a jousting helm from the 16th century - kind of a different beast.

Finally, how were these garments made? It seems that in many cases they were made more locally than plate armours. In the Howard accounts, armours are bought finished and ready to wear. However, often cloth is bought for doublets of fence, implying that they were constructed by people employed in Howard's (very large) household. However, fashionable garments for the very wealthy (which describe most surviving quilted medieval garments) might be made by specialists.