This arcane-sounding term refers, not to "an impartial cloth", but to the cloth that is partial to the surface of the altar which, here at Trinity we call, informally, the Altar Cloth.

However, whatever name it's given, it is, essentially, a table cloth, albeit the one upon which the fare of our salvation is laid.

We Christians are, if nothing else, assiduous about our "table" manners and the insistence on the altar being covered with a fair linen illustrates this sentiment.

In fact, in the Book of Common Prayer (p. 67), the very first rubric in the Communion Service is:  "The Lord's Table, at the Communion time, shall have a fair white linen cloth upon it."

Likewise the Eucharist in the Book of Alternative Services has as its first instruction (p. 183),   "The holy table is spread with a clean (white) cloth during the celebration."

Hence, not only is it considered essential form to have an appropriate covering for our Lord's table, it is important that "the fair linen" be "fair" i.e., both "clean" and "white".

Moreover, in keeping with good "table" manners, in earlier days the BCP rubric was followed to the letter, i.e., the fair linen was not spread upon the altar until the Offertory (this custom has stood the test of time, and is practised still in a few Anglican churches)

However, whether its spread immediately before the meal or just before the guests arrive, the fair linen is often more opulent than your run of the (textile) mill table cloth.

For example, most fair linens are embroidered (always white); often with five crosses:   one in the middle, and two at each end (symbolising the five wounds of Christ).

As for materials, linen continues to be almost exclusively the fabric of choice, but it will noted that the BAS makes discreet allowance for something else.   (Fair linens are very expensive - close to $1000, and as a former rector of mine once quipped "they wrinkle if you so much as look at them" )

At the end of the day the fair linen's material is immaterial.

What matters is that we select and care for fair linens with an appropriate measure of reverence; in this way we offer to God our best, for he has offered us nothing less!