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," which appears as "ﬁ".
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.
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. 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 moderately rich set of ligatures is the Computer Modern Roman typeface that is provided with TeX, which includes the ff, fi, fl, ffi, and ffl ligatures which TeX automatically uses when it finds these letters juxtaposed in the text.
This table shows unligatured sets of letters on the left, and the corresponding Unicode ligature on the right. Not all browser/operating system combinations will render the table correctly.
It is important to note that the letter Æ (æ) when used in the Danish or Norwegian languages is not a typographical ligature, and must never be treated as such. It is a distinct letter and vowel. Likewise, the letter ß is a distinct letter in the German, and the letter ĳ/Ĳ is a distinct letter in Dutch.