Blackletter
Blackletter is the earliest printed type, and is base on hand-copied texts. It is traditionally associated with medieval German and English (Old English), but has recently seen more use. Blackletter was revived as a ‘pure German’ form in Nazi Germany, and is extensively used by (particularly) Latino gangs as implying officialness or deep seriousness. Blackletter dates from around 1450.
Oldstyle
Oldstyle has uppercase letterforms based on Roman inscriptions, and lowercase based on Italian humanist book copying. It is typified by a gradual thick-to-thin stroke, gracefully bracketed serifs, and slanted stress, as indicted by the red line through the uppercase ‘O’, and as measured through the thinnest parts of a letterform. It remains one of the most readable classes for text, due to the moderate stroke variations and good distinction between letterforms. Oldstyle dates from around 1475.
Italic
Usually considered a component of the roman family of a font, italic really deserves its own class. Based on Renaissance Italian Humanist handwriting, italics are casual as opposed to the more formal roman forms of a font. Italics are generally used for emphasis, captions, and the like, and not for body text. It is important to remember to use true italics as opposed to digitally generated versions. Italics for sans-serif (and occasionally other) fonts are often called obliques. Date from around 1500.
Script
As mentioned above, oftentimes anything seemingly based on handwriting is lumped under script. To be more precise, script is a formal replication of calligraphy. Script may also be based on engraved forms. As type, script is unsuitable for text, but is widely used to lend a formal element to a layout. Dates from 1550.
Transitional
As the name implies, transitional bridges the gap between oldstyle and modern. Largely due to technological advances in casting type and printing, transitional embodies greater thick-to-thin strokes, and smaller brackets on serifs. Stress moves to be more vertical. Dates around 1750.
Modern
Furthering the trends started with transitional, modern pushes to extreme thick-to-thin strokes, and unbracketed (square) serifs. Many modern typefaces lose readability if set too tight, or at too small a size, particularly with strong vertical stress. Dates from 1775.
Slab (Square) Serif
Slab or square serif was developed for heavy type in advertising. Also known as Egyptian (it appeared during the Egyptology craze in Europe), slab serif generally has little variation in stroke weight: it's generally uniformly heavy. Also with slab serif, letterforms are becoming more geometric, and less calligraphic. Dates from 1825.
Sans Serif
Although appearing earlier, sans serif gained much popularity in the twentieth century, mainly as a move towards an international aesthetic in typography. San serif can be strictly geometric, as in Futura, or more humanist, as with Gill Sans. More recently, sans serifs with a variation in stroke weight are becoming more common (Optima, Myriad). Dates from 1900 in common use.
Serif/sans serif
A fairly recent development are families of typefaces with both serif and sans serif fonts. These provide the designer with even more unified variation than an extensive family of serif or sans serif. Dates from 1990.