First off, an important note: the terms typeface and font are not synonymous. The typeface Raleway is available in a variety of styles (italic, condensed, etc.) and weights (light, book, bold, etc.) but only becomes a font when the typeface, size, style and weight are all specified simultaneously.
A sample of the styles and weights available for the Raleway typeface.
Blackletter was the first typeface, created by Johannes Gutenberg, the inventor of the printing press. It was intended to replicate the hand-scribed appearance of the books that existed before Gutenberg’s printing press was invented.
Old Style serif typefaces are characterised by thick serifs and low contrast between the thick and thin strokes used to create glyphs (i.e. there is not much difference in thickness between the thin and thick strokes). They are diagonally stressed, meaning that the thinnest parts of the glyphs are at an angle.
Transitional serif typefaces are characterised by thinner serifs and higher contrast between glyphs’ thick and thin strokes.
Modern serif typefaces are characterised by very fine and long serifs and very high contrast between glyphs’ thick and thin strokes. They are vertically stressed, and have thick vertical lines.
Slab Serif typefaces are characterised by extremely thick, block-like serifs. They generally have no brackets (the curved features that connect strokes to serifs).
Grotesque sans serif typefaces have squarer curves and only minor variation in stroke widths. They were the earliest of the sans serif typefaces, with later designs becoming neo-grotesque.
Neo-grotesque sans serif typefaces lie somewhere between the geometric and humanist sans serifs (they are sometimes referred to as transitional) that followed, with simple shapes but greater variation in stroke width.
Geometric sans serif typefaces are based on simple geometric shapes like circles and squares and have very little contrast in stroke-width.
Humanist sans serif typefaces are characterised by the presence of gentler, more natural curves than those seen in geometric sans serif fonts and have greater variation in stroke width.
Pixel Type fonts were originally invented for low-resolution displays, and are only commonly used now when space (in pixel terms) is restricted.
Script fonts are designed to reproduce handwriting or calligraphy.
Display fonts are designed for use in displays, and vary incredibly widely in their design. Common display fonts attempt to reproduce LCD or dot-matrix displays, appear carved, stencilled or stamped, etc.
And to close, a lovely stop-motion video on The History of Type, by Ben Barrett-Forrest:
* Computer Modern has preferred status amongst physicists, as it is the default font of the LaTeX typesetting system.