In order to use PostScript fonts, TeX needs metric (called TFM) files. Several sets of metrics are available from the archives; for mechanisms for generating new ones, see metrics for PostScript fonts. You also need the fonts themselves; PostScript printers come with a set of fonts built in, but to extend your repertoire you almost invariably need to buy from one of the many commercial font vendors (see, for example, "choice of fonts").
If you use LaTeX2e, the best way to get PostScript fonts into your
document is to use the PSNFSS package maintained by Walter
Schmidt; it's supported by
the LaTeX3 project team, so bug reports can and should be submitted.
PSNFSS gives you a set of packages for changing the default
roman, sans-serif and typewriter fonts; e.g., times.sty will set
up Times Roman, Helvetica and Courier in place of Computer Modern,
while avant.sty just changes the sans-serif family to
AvantGarde. To go with these packages, you will need the font metric
files (watch out for encoding problems! - see
metrics for PostScript fonts)
and font description (.fd) files for each font family you
want to use. Many sets of metrics, etc., can be obtained from the
psfonts area of CTAN, arranged by
vendor (e.g., Adobe, Monotype, etc.). For convenience,
metrics for the common '35' PostScript fonts found in most printers
are provided with PSNFSS, packaged as the "Laserwriter set".
For older versions of LaTeX there are various schemes, of which the simplest to use is probably the PSLaTeX macros distributed with dvips.
For Plain TeX, you load whatever fonts you like; if the encoding of the fonts is not the same as Computer Modern it will be up to you to redefine various macros and accents, or you can use the font re-encoding mechanisms available in many drivers and in ps2pk and afm2tfm.
Victor Eijkhout's Lollipop package supports declaration of font families and styles in a similar way to LaTeX's NFSS, and so is easy to use with PostScript fonts.
Some common problems encountered are discussed elsewhere (see problems with PS fonts).