Cotton is a cellulose fibre of vegetable origin. It has a fairly long staple and can thus be spun into very fine thread. It is hard-wearing and takes colour well. However it was not grown in Europe in period, but was imported from the Middle East. The fibres of cotton itself are shorter than linen, making it a poor choice for a warp and conseuqently tends to make a coarse cloth.
"Arab merchants brought cotton cloth to Europe about 800 A.D. When Columbus discovered America in 1492, he found cotton growing in the Bahama Islands. By 1500, cotton was known generally throughout the world".
The word cotton originally referred to a weave or finish rather than a fibre or cloth.
Cotton (as a cloth rather than a finish) wasn't known in England until after 1300 where it initailly pops up as a surgical dressing (and very expensive one at that).
The word fustian has been around since the 12th century (England) and now refers to "a hard-wearing fabric of cotton mixed with flax or wool with a slight nap" although it appears to have originally referred to a linen and wool cloth.