Source: edited from Wikipedia Typeface
Typeface classification is not easy because typographic tradition is self-referential – type designers often use older forms as a basis for experimentation and innovation.
Typeface design is also related to technology. From early Roman to 15th Century Western type was drawn by hand, either with a brush, a flat reed pen or a chisel. From mid-1500s casting letters in lead allowed new precision in form. In the 20th century use of cheap mass produced paper stock for eg newspapers and telephone directories meant that legibility had to allow for some bleeding of ink. Digital technology and type design software in the last decade has led to a rapid proliferation of typefaces by individual designers and artists.
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.
20th Century classification systems
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.
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), Égyptiennes (slab serifs), Didots and Elzévirs (faces with triangular serifs).
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.
Currently the primary “official” classification system is the Vox-ATypI system originally put together in 1954 by Maxmilien Vox. Originally a ten-part classification, Vox revised his original proposal within months to a more compact nine-part scheme. These were subsequently expanded to 11 general categories, including non-Western type, with some subdivision. This classification tends to group typefaces according to their main characteristics, often typical of a particular century (15th, 16th, 17th, 18th, 19th, 20th century), based on a number of formal criteria: downstroke and upstroke, forms of serifs, stroke axis, x-height, etc ???. Although the Vox-ATypI classification defines archetypes of typefaces, many typefaces can exhibit the characteristics of more than one class.
The Vox-ATypI system was adopted in 1962 by the Association Typographique Internationale (ATypI), and in 1967 as the basis for British Standards Classification of Typefaces (BS 2961:1967). But at the 2010 ATypI general meeting, the association stated that the Vox-ATypI system was seriously flawed, and to create a new working group on typeface classification.
Adobe, Wikipedia and the growing numbers of font websites also have their own classifications.
Serif fonts
Classical or old style
characterized by triangular serifs, oblique axis and low stroke contrast
Humanist or Venetian

- short and thick bracketed serifs, ascenders with slanted serifs
- stress that approximates that of a broad-nibbed pen held at an angle to the page
- a slanted cross stroke on the lowercase ‘e’,
- low contrast between horizontals and verticals and thick and thin strokes.
Examples: Centaur, Adobe Jenson, Berkeley Old Style and Cloister.
Garalde

- finer proportions than the humanists
- horizontal cross-stroke on the lower case e
- stronger contrast between downstroke and upstroke, thick and thin strokes.
- stress is inclined on an oblique axis to the left
- bracketed serifs.
Transitional

- vertical, or near vertical, stress
- marked contrast between main and connecting hairline strokes
- serifs are thin, flat and bracketed
Examples: Baskerville, Times Roman, Perpetua, Stone Serif.
Serif Fonts: Moderns
The moderns can be broken down into Didone, mechanistic and linear categories, and are characterized by a simple, functional feel that gained momentum during the industrial period.
Didone or Modern

- very strong contrast between full and connecting strokes (the connecting strokes being extremely fine
- vertical stress
- unbracketed hairline serifs
- constructed rather than drawn.
Examples: Bodoni, Didot, Fenice and Walbaum.
Serif fonts: Mechanistic, slab or Egyptian

- sturdy
- very low contrast in stroke weight
- includes both typefaces with bracketed serifs (clarendons or ionics) and typefaces with heavy square/rectangular or unbracketed serifs (egyptians).
Sans Serif or Lineal
Lineals, or linéales, combine all typefaces without serifs (called sans-serif, gothic, or grotesque), all of which correspond to the Antiques of the Thibaudeau classification. But they were not widely adopted until the end of the 19th Century. The British Standard 2961 broke this group into 4 subcategories: Grotesque, Neo-Grotesque, Geometric, and Humanist.
Grotesque

- some contrast between thick and thin strokes
- terminals of curves are usually horizontal
- frequently has a spurred “G” and an “R” with a curled leg.
Examples: Franklin Gothic, Bell Gothic and Bell Centennial, Frutiger, Headline, Monotype 215, and Grot no. 6.
Neo-grotesque or realist

- they generally do not have a spurred “G”
- terminals of curves are usually slanted
- often have a large degree of subtlety and variation of widths and weights to accommodate different means of production (Hot type, foundry type, phototypesetting, see History of typography, 20th century).
Examples: Helvetica and Univers.
Geometric
Examples: Eurostile and Futura.
Humanist

Examples: Gill Sans and Optima.
Calligraphics
The Calligraphics can be broken down into glyphic, script, graphic, blackletter, and Gaelic categories, and are characterized by a suggestion of being hand-crafted.
Glyphic

- small, triangular serifs or tapering downstrokes.
- greater emphasis on the capital letters in glyphic typefaces, with some faces not containing a lowercase.
Examples: Albertus, Copperplate Gothic, and Trajan.
Script

- seem to be written with a quill, and have a strong slope.
- letters can often be connected to each other.
Examples: Bickham Script Pro, Champion Script Pro, Raniscript, Shelley, Mistral and Francesca.
Graphic
Examples: Banco and Klang.
Blackletter

- pointed and angular forms
- modelled on late medieval hands written with a broad-nibbed pen.
Gaelic
Gaelic type was added to the classification at the AGM of the Dublin meeting of ATypI, on 12 September 2010.
Non-Latin or ‘exotics’
Gathers (without distinction of style) all writing systems not based on the Latin alphabet: Greek, Cyrillic, Hebrew, Arabic, Chinese, etc. English printers traditionally called these exotics.
Websites and font libraries
Font recognition
identifont: http://www.identifont.com
Free fonts
http://www.creativebloq.com/graphic-design-tips/best-free-fonts-for-designers-1233380
http://ilovetypography.com/download-free-fonts/
http://www.awwwards.com/the-100-greatest-free-fonts-for-2014.html
Specialist
Generates type pictures from photos
http://www.typoeffects.com/?page_id=135