A leaf is a single complete page, front and back, in a finished book. The recto side of a leaf faces left when the leaf is held straight up from the spine. The verso side of a leaf faces right when the leaf is held straight up from the spine.
A folio is a single sheet folded in half to make two leaves. The term folio can also be used in the same sense as leaf.
A codex is a set of folios nested together and sewn through the fold.
A signature is a large sheet printed with several pages, intended to form four or more leaves in the finished book. The pages are arranged on the sheet so that all of the pages orient the same way and are in proper sequence after the sheet is folded. Arranging these pages correctly is called imposition. Signature also refers to a sequence number or code printed on the sheet so that the several signatures that makes a complete book may be properly sequenced; this signature is often trimmed off after binding. The signature may be folded in several ways, depending on the number of leaves it will form. It is then stitched together down the last fold.
A sheet folded in quarto is folded in half twice at right angles to make four leaves. A sheet folded in octavo is folded in half 3 times to make 8 leaves. A sheet folded in sextodecimo is folded in half 4 times to make 16 leaves.
A quire is a set of leaves that are stitched together. This is most often a single signature, but may be several nested signatures. The quires for a single book are arranged in order and then stitched together as a set.
Trimming allows the leaves of the bound book to be turned. A sheet folded in quarto will have folds at the spine and also across the top, so the top folds must be trimmed away before the leaves can be turned. A signature folded in octavo or greater may also require that the other two sides be trimmed.
Resources:
WWW.ENWIKIPEDIA.ORG
WWW.PALIMPSEST.STANFORD.EDU
WWW.E-BOOKBINDING.COM