lementary classification

The oldest classification system is due to Thibaudeau in the beginning the 20th century. The typefaces are classified according the form of their serifs. Four different cases can be in fact easily identified:
 

Elzeveir	Didot	Egyptian	Antique
Triangular serifs.	Horizontal and thin serifs.	Weight of serifs identical as the strike.	Sans serifs.
Palatino	Bodoni	Glypha	Optima
Garald  & Transitional	Didone	Mechanistic	Lineal

raditional roman typefaces

It is also possible to classify the roman typefaces on an historical base. The Garaldes are based on drawings of the XVth and XVIth  centuries, the Reales on drawings of the XVIIth and XVIIIth, the Didones on drawings of the XVIIIth and XIXth and finally the Mecanes on drawings of the XIXth.
 

Garald	Transitional	Didone	Mechanistic
Lowercase height is smaller compared with capitals.	Lowercase height is bigger than for Garalds.	Lowercase height is small compared with capitals.	Lowercase height is bigger than for Garalds.
Bembo	Times	Walbaum	Clarendon
Relatively uniform stroke widths.	Greater varia- tion between the vertical and horizontal stress.	Strong contrast between thick and thin stroke.	Little contrast between thick and thin stroke.
Garamond	Charter	Modern 216	Bookman
The axe of the letters is oblique.	The axe of the letters is slightly oblique.	The axe of the letters is vertical.	The axe of the letters is vertical.
Galliard	Baskerville	URW Antiqua	Century

ans serif typefaces (lineals)

Sans serif typefaces used today are all born in the 20th century. The contrast between thick and thin strokes of these types is often small. Moreover, the lack of serifs makes them less legible. Anyway, on a typographical point of view, they are the incarnation of modern typogaphy and are also widely used.
 

Classical	Humanist	Geometric
Light contrast between thick and thin stroke.	Contrast between thick and thin stroke similar as it is for the roman types.	No contrast between thick and thin stroke.
Letter Gothic	Gill Sans	Avant Garde
Large x-height.	Classical x-height (cf roman typfaces).	Large x-height.
Franklin Gothic	Optima	Futura
Uniform design.	The ’g’ is often closed. Letters are closed from their roman counterparts.	The ’a’ is often designed as a loop. Letters  are inspired by geometry.
Helvetica	Syntax	Avant Garde

ther categories (miscelleanous)

This category usually applies to any typeface not fitting into any of the previous categories. They are not very legible but are unique and highly distinctive.
 

American Uncial	Uncial: typefaces inspired by the calligraphic writings of the 5th century in which capital and minuscule letters are mixed.
Fette Fraktur	Fraktur: typefaces inspired by the calligraphic writings of the Middle-Age. Not very legible for long texts.
Vivante	Script: calligraphic writing, not always very legible.
Arnold Boecklin	Art Nouveau: Inspired by the modern style (end of the 19th, beginning of the 20th).
Stop	Futurist: typefaces created in the 60’s-70’s, inspired by the Science Fiction movement.
Caslon Antique	Pop art/Grunge: contemporary distorted typefaces.
VAG Rounded	Novelty: Unclassified typefaces.
Zapf Dingbats	Dingbats: Symbols typeface.