ASCII (an acronym for American Standard Code for Information Interchange) is an encoding for various glyphs (written forms such as letters, numbers, etc) into 7-bit numeric form.
Originally based on the English alphabet, the supported set for ASCII includes not only numbers, letters (upper- and lower-case) but also punctuation, and other special-purpose characters (from then-common ones like '@', '#', etc, to others that ASCII has made popular - '^', '|', etc). It also includes non-printing characters used for control of printing terminals - tab, line feed, carriage return, etc.
ASCII was developed in the early 1960s from telegraph code. Its first commercial use, in 1963, was as a teleprinter code promoted by Bell data services for the TWX network. Later, it was adopted for use by terminals, line-printers, etc, as well as by secondary storage and files.
ASCII soon superseded an earlier wide-spread encoding, SIXBIT, which allowed 6 characters to be carried in the then-common 36-bit words common on many computers (but only supported upper-case characters). IBM had its own competing encoding standard at the time, EBCDIC, which ASCII has also gradually superseded.
ASCII was eventually used by personal computers such as the Apple II, and the IBM PC. IBM extended the character set with a separate set of glyphs used for line drawing; the first of many often-incompatible extensions to ASCII for various national, etc, usages. It has now been superseded for the WWW by UTF-8, which is however backwards compatible with ASCII.