ASCII Art is art that is created entirely with text characters. In its purest form, it is created entirely with the printable characters of the ASCII character set, but the term has sometimes been used more loosely to refer to other character art which might make use of "extended" character sets or added control or escape codes. When ANSI escape codes are used, that is known as ANSI Art.
Predecessors existed even before ASCII was first defined in 1963; people were using typewriters to create art as early as the 1800s, and teletypes in the 1920s were also used for artwork. When computers began to appear, it became a common recreation to come up with pictures or cartoons printed entirely in text characters (such as the Snoopy picture ubiquitously found on the walls of computer centers). The BBS world of the 1970s through 1990s also used a lot of text art, though it often used platform-specific character sets which included things like line-drawing characters, so it wasn't strictly ASCII art. The rise of fully-graphical computers has reduced the frequency of ASCII art use, but it still turns up from time to time.
Pure ASCII art uses only printable characters, but there have also been instances of control characters being used to do special effects for artistic purposes on terminals (screen and printing) that supported particular functions such as overstrike and animation effects using backspaces or carriage returns without linefeeds. These don't work consistently on all systems, however.
In general, ASCII art (and its variants for other character encodings) is done using fixed-width typewriter-style characters, expected to be viewed in a font such as Courier in which every character is the same width and hence everything lines up properly. When viewed in a variable-width font (such as most computer fonts are these days), the picture is likely to be messed up in appearance.