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In principle, TeX can use any font for which it has metrics (character widths, kerning, etc.), and shapes (these days, generally PostScript Type 1 or TrueType or OpenType).
A couple notable articles on basic font concepts and usage in LaTeX:
For additional information of all kinds:
Font licensing is a perennial and troublesome topic. The freely available fonts listed below are under a variety of licenses; some commonly-used licenses for fonts are listed separately.
Notable font utilities include:
Computer Modern: Getting to specifics, the first and still most prevalent typeface in the TeX world is Computer Modern, developed by Donald Knuth using his novel Metafont program, which is distributed alongside TeX.
Other Modern: Variants of Computer Modern have been created which cover letters and symbols from virtually all other Latin-based languages, as well translating the characters into PostScript Type 1 fonts, etc. Latin Modern, cm-super, and NewComputerModern (also TUGboat article on NewCM).
Base PostScript: TeX also comes with support for the 35 standard PostScript fonts: Times Roman, Helvetica, Courier, Palatino, et al. (URW++ graciously donated these fonts to the public in 1996, thanks to the efforts of Peter Deutsch for Ghostscript. In 2009, URW also made them available under the LPPL, thanks to the GUST font team's request. The GPL'd originals, are available for reference. Version 4.0 of the fonts was the basis for subsequent work by GUST. The current versions distributed by Ghostscript are substantially different.
Please see the psnfss2e documentation for usage in LaTeX, and the test1.tex (and the other test*.tex) for more examples of output and use. In brief:
\usepackage{mathptmx} % times roman, including math (where possible)
\usepackage{mathpazo} % palatino, including math (where possible)
\usepackage{helvet} % helvetica
The TeX Gyre fonts from the GUST Typefoundry extend the base 35 PostScript fonts to many other scripts, analogous to Latin Modern's extensions of Computer Modern.
The fonts below cover typesetting in a wide variety of languages and scripts, as well as of most mathematics. Font sampler documents are available, showing what they look like:
These fonts come with standard TeX installations, such as TeX Live and MiKTeX, ready to use.
These free (libre) fonts do not have Type 1 versions and therefore are most easily used with XeTeX and LuaTeX. (Some of the fonts above and below also have OpenType and/or TrueType versions.) They are all included in the principal TeX distributions.
While we're here, the article OpenType Math Illuminated by Ulrik Vieth gives a detailed comparison of OpenType and TeX math font parameters.
A few additional typefaces are available for general use by individuals, but have various restrictions which prevent inclusion in the free TeX distributions:
The getnonfreefonts script simplifies installation of these fonts, and others. Try getnonfreefonts --help.
First, if you are interested in helping produce more free fonts, please see the GNU Font Utilities web page and this TUGboat article for starting points, and let us know.
Excellent fonts are available commercially from many companies. Unfortunately, the ones we checked (Adobe, Bitstream, et al.) have extremely high prices for a font library (thousands of dollars). Prices for individual typefaces are also expensive. There are a number of inexpensive collections available, of varying quality and legality, but since none are legally usable everywhere in the world, regrettably TUG cannot specifically recommend any of them. Commercial fonts typically do not come with support for TeX, such as .tfm files, but many such metrics are available on CTAN; see especially Walter Schmidt's collection and MinionPro TeX support.
One commercial font collection we can recommend is MathTime Pro 2 by Michael Spivak, distributed by Personal TeX, Inc. It has a full complement of Times Roman characters for math typesetting with TeX, based on original designs. It can be used with any TeX implementation that supports TeX's virtual fonts.
Finally, the Lucida typeface family, while remaining under a proprietary license, is available through TUG, in both Type 1 and OpenType versions.