A quire of paper is today used as a measure of paper quantity. The usual meaning is a set of 24 or 25 sheets of paper of the same size and quality. It might also be thought of as 1/20 of a ream.
It originally had other meanings:
A quire (also called a “gathering”) was in the Middle Ages most often formed of 4 folded sheets of vellum or parchment, i.e. 8 leaves, 16 pages. The terms “quaternion” (or sometimes quaternum) designate such a quire. A quire made of a single folded sheet (i.e. 2 leaves, 4 pages) is a “bifolium” (plural “bifolia”); a “binion” is a quire of two sheets (i.e. 4 leaves, 8 pages); and a “quinion” is five sheets (10 leaves, 20 pages).
The current word “quire” was derived when quaternum was shortened to “quair” or “guaer” in common usage. Afterwards, when bookmaking switched to using paper and it became possible to easily stitch 5 to 7 sheets at a time, the association of “quaire” with “four” was quickly lost.
It also became the name for any booklet small enough to be made from a single quire of paper. Simon Winchester, in The Surgeon of Crowthorne, cites a specific number, defining quire as “a booklet eight pages thick.”
In blankbook binding, quire is a term indicating 80 pages.