A signature, a gathering, a quire
Three words for nearly the same thing, and the small differences that still matter at the bindery.
A quire is a set of sheets folded together and nested one inside another, ready to be sewn. It is one of the oldest units of bookmaking, older than printing, and it is why the pages of a hardback come in clumps rather than as loose leaves.
A gathering is the same object, described from the binder's side of the bench. A signature is strictly the mark — a letter or number printed in the tail margin of the first page of each gathering, so the person collating them can see at a glance that gathering H follows G. The word slid across to mean the gathering itself, and now all three are used interchangeably by almost everyone.
Why the distinction is not pedantry
Because the signature mark is the only one of the three you can point at on a finished book. Open an older hardback at the spine and look at the bottom of the first page of each clump: the small letters are still there, doing the job they did in 1490.
Four leaves, eight pages
The commonest quire is four sheets folded once: eight leaves, sixteen pages. Hence quaternion, hence the Latin quaterni, four together, which is where the word itself comes from.
It is a good word for a thing made of folded sheets. That is what a book is, and it is what this software is named after.