A ligature occurs where two or more letterforms are written or printed as a unit. Generally, ligatures replace characters that occur next to each other when they share common components. Ligatures are a subset of a more general class of figures called ``contextual forms.'' Contextual forms describe the case where the particular shape of a letter depends on its context (surrounding letters, whether or not it's at the end of a line, etc.).
One of the most common ligatures is ``fi''. Since the dot above a lowercase 'I' interferes with the loop on the lowercase 'F', when 'f' and 'i' are printed next to each other, they are combined into a single figure with the dot absorbed into the 'f'.
An example of a more general contextual form is the greek lowercase sigma. When typesetting greek, the selection of which 'sigma' to use is determined by whether or not the letter occurs at the end of the word (i.e., the final position in the word).
Ligatures were originally used by medieval scribes to conserve space and increase writing speed. A 14th century manuscript, for example, will include hundreds of ligatures (this is also where ``accents'' came from). Early typefaces used ligatures in order to emulate the appearance of hand-lettered manuscripts. As typesetting became more automated, most of these ligatures fell out of common use. It is only recently that computer based typesetting has encouraged people to start using them again (although 'fine art' printers have used them all along). Generally, ligatures work best in typefaces which are derived from calligraphic letterforms. Also useful are contextual forms, such as swash capitals, terminal characters, and so on.
A good example of a computer typeface with a rich set of ligatures is Adobe Caslon (including Adobe Caslon Expert). It includes:
Upper case, lower case, small caps, lining numerals, oldstyle numerals, vulgar fractions, superior and inferior numerals, swash italic caps, ornaments, long s, and the following ligatures:
ff fi fl ffi ffl Rp ct st Sh Si Sl SS St (where S=long s)
[Ed: Another common example is the Computer Modern Roman typeface that is provided with TeX. this family of fonts include the ff, fi, fl, ffi, and ffl ligatures which TeX automatically uses when it finds these letters juxtaposed in the text.]
While there are a large number number of possible ligatures, generally only the most common ones are actually provided. In part, this is because the presence of too many alternate forms starts reducing legibility. A case in point is Luxeuil Miniscule, a highly-ligatured medieval document hand which is completely illegible to the untrained eye (and none too legible to the trained eye, either :)).
Ligatures were used in lead type, originally in imitation of calligraphic actions (particularly in Greek which retained an excessive number of ligatures in printed material as late as the 19th century), but as typefaces developed, ligatures were retained to improve the appearance of certain letter combinations. In some cases, it was used to allow certain letter combinations to be more closely spaced (e.g., ``To'' or ``Vo'') and were referred to as ``logotypes''. In other cases, the designs of two letters were merged to keep the overall spacing of words uniform. Ligatures are provided in most contemporary fonts for exactly this reason.
The term ligature should only be used to describe joined letters in printing, not letters that overlap in manuscripts.
Many (not all) accents came from the practice of using a tilde or other mark to represent an omitted letter, so that for example the Latin word `Dominus' would be written dns, with a tilde or bar over the n. This is an abbreviation, not a ligature.
Most ligatures vanished during the 15th and 16th Centuries. It was simply too much work to use them, and it increased the price of book production too much.
[Ed: there is no ``complete'' set of ligatures.]