Adobe has specified a number of formats for files to represent fonts in PostScript files; this question doesn't attempt to be encyclopaedic, but we'll discuss the two formats most commonly encountered in the (La)TeX context, types 1 and 3.
Adobe Type 1 format specifies a means to represent outlines of the glyphs in a font. The 'language' used is closely restricted, to ensure that the font is rendered as quickly as possible. (Or rather, as quickly as possible with Adobe's technology at the time the specification was written: the structure could well be different if it were specified now.) The format has long been the basis of the digital type-foundry business, though things are showing signs of change.
In the (La)TeX context, Type 1 fonts are extremely important. Apart from their simple availability (there are thousands of commercial Type 1 text fonts around), the commonest reader for PDF files has long (in effect) insisted on their use (see PDF quality).
Type 3 fonts have a more forgiving specification. A wide range of
PostScript operators is permissible, including bitmaps operators. Type 3
is therefore the natural format to be used for programs such as
dvips when they auto-generate something to represent
Metafont-generated fonts in a PostScript file. It's Adobe Acrobat Viewer's
treatment of bitmap Type 3 fonts that has made direct Metafont output
inreasingly unattractive, in recent years. If you have a PDF
document in which the text looks fuzzy and uneven in Acrobat Reader,
ask Reader for the File->
Document Properties->
Fonts ..., and it will show some font or other as "Type 3"
(usually with encoding "Custom"). (This problem has disappeared
with version 6 of Acrobat Reader.)
Type 3 fonts should not entirely be dismissed, however. Acrobat Reader's failure with them is entirely derived from its failure to use the anti-aliasing techniques common in TeX-ware. Choose a different set of PostScript graphical operators, and you can make pleasing Type 3 fonts that don't "annoy" Reader. For example, you may not change colour within a Type 1 font glyph, but there's no such restriction on a Type 3 font, which opens opportunities for some startling effects.
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