Quite simply, a ligature is the combination of two letterforms into one continuous character. For many people, the most familiar ligature is the ampersand (&) or and sign. This character is the stylized abbreviation of "et", the Latin word for "and". The word "ampersand" is derived from the phrase "and per se and". Adobe Systems has an overview of this particular character at its site.
Another common (alleged) ligature is the "at" symbol (@), which has becomely widely used with the advent of e-mail. Like the ampersand, this character's origins seem to be derived from Latin, specifically the word "ad", meaning, logically enough, "at". The Terminal article, Where it's @ has more information on this character.
Below are some of the ligatures appearing in the standard Western ASCII set. You may see these used in words such as 'Æsthetic', 'Encyclopædia' and 'œuvre'

In 1959 Herbert Bayer designed his Fonetik Alfabet as a proposed simplification of the existing Roman alphabet. This system of writing sought to eliminate the endless variables found in written language and assign one sound to one letter. Within this system, double letter and triple letter sounds were also simplified using ligatures, as illustrated below. 

Even ancient Egyptians sought to simplify common combinations of symbols in Hieroglyphic ligatures. There are hundreds of examples of this type of pictographic shorthand. Here is just one example of this type of Picto-ligature.
The crowns of Upper and Lower Egypt each represent their respective kingdoms. The combined crown represents a united Egypt. The latest (quasi) ligature to be forced onto the world is the Euro, the symbol for the United European currency.
by Richard Kegler

 

The Ampersand
Max Caflisch, Adobe Systems

 

Where it's @

 
PDF with full ASCII character set offered in most P22 fonts
 
 
Daniel Will-Harris gives his Ligature Schmigature arguments against the need for any ligature in the English language. Many specific visual examples are given for a clear explanation about the general aesthetics of ligatures.

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