Extended character is another name for a "special character" or "modified letter" outside a traditional alphabet or character set. The name refers to characters that are additional to an original character set.
The presence of such characters posed challenges for typing these languages before computers, and later for using these languages on computers. Early efforts to accommodate these orthographies on computers involved creating "special fonts" based on 8-bit encodiing, which were of limited use for exchanging digital documents or developing web content. Unicode in principle provides for encoding of extended characters and diacritics in a way that overcomes these limitations.
Most discussion of extended characters refer to extended Latin characters. There are also extended Arabic characters (some of those are useful for Ajami writing in Africa) and extended Ge'ez/Ethiopic characters.
Some years ago, when 7-bit ASCII fonts were being replaced by 8-bit fonts, the latter were sometimes referred to as "extended ASCII," and the upper range of characters as "extended characters." (For references to this, see for example "Extended Characters and MPE").