ASCII art is any kind of artwork: pictures, charts, cartoons, whatever... drawn with the characters in the ASCII character set.
The ASCII (American Standard Code for Information Interchange) character set is a set of 128 characters (0 to 127) which are standard on almost all types of computer. The only characters used in ASCII art are those with the values 32 to 126 [-], and 13, which represents a carriage return (new line).
The other characters in the ASCII character set (0-12, 13-31, and 127) are control codes for representing things such as "end of file" and "backspace"; they should not be used in ASCII art.
It is the most universal computer art form in the world -- every computer system capable of displaying multi-line text can display ASCII art, without needing to have a graphics mode or support a particular graphics file format.
From the Ascii art FAQ by Matthew Thomas: