Over the past century, quite a few classification systems have been proposed. Most are generally believed to be subjective and incomplete, and many of them use the same terms for similar but slightly different classes. The impossibility of a truly complete classification system has led many people to dismiss any attempt to classify typefaces ? there are simply too many variables to make anything close to a practical, comprehensive system. Essentially, classification describes typefaces; it does not define them. It?s not inflexible, and is more of an aid than a rule. However, for the reasons given above, I believe there is value to be found in it. Below are a few examples.
The primary ?official? classification system currently is the Vox-ATypI system. Originally put together in 1954 by Maxmilien Vox, it was adopted in 1962 by the Association Typographique Internationale (ATypI), which made a minor change at the 2010 conference (appropriately, held in Dublin) to include Gaelic as an extra category. It classifies typefaces in 11 general categories, with some subdivision. Its Wikipedia article provides an excellent overview.

Bringhurst, in his Elements of Typographic Style ? perhaps the standard in typographic textbooks today ? categorizes typefaces loosely after periods of art history; for example, Baroque, Rococo, Romantic, etc. A book designer himself, Bringhurst focuses on text typefaces and practically ignores display type.
Others are much more general. An early system by French typographer Francis Thibaudeau, which provided the base for Vox?s later more thorough classification, includes four broad categories: Antiques (sans serifs), ?gyptiennes (slab serifs), Didots and Elz?virs (faces with triangular serifs).
Gerrit Noordzij, while at the Royal Academy of Fine Arts in the Hague, held that typography was essentially an extension of handwriting, teaching typography using loose categories of letters that might be written with a broad-nib or pointed-nib pen, as well as interrupted or uninterrupted strokes, with varieties of both serifs and sans falling into each category.
These are just a few of the ways people have classified type over the years. In this two-part article, I will condense the various methods slightly and present what is at the very least generally accepted as legitimate (as there will always be a few out there who refuse to give up a particularly unusual classification method, or who decry any method at all).