Originally a semi-derisive term for Japanese, used whenever things like scans of Japanese magazines (fairly common in /a/, /m/, and /v/) are posted, typically to ask for a translation of the moonspeak. The term was first coined in /b/, when someone posted a picture from the Tick comic book where the hero is fighting ninjas speaking in Japanese characters, and he tells them he doesn't understand their moon language. Thus, languages like Japanese were initially called "moon language," but this failed catch on and now 'moonspeak' is the generally accepted term. Its use has since expanded to mean any language that isn't English, especially if it uses lots of special characters (like umlauts) or, as with Japanese, a different character set entirely (which is even worse if you don't have the fonts installed). The most common examples on /b/ are Scandinavian languages like Swedish and Finnish, owing to the relatively large number of /b/tards in those countries; other European languages have also been called moonspeak, usually German and Russian. Korean has also shown up on /b/ from time to time, usually in pictures. Numberous threads of "true" moonspeak (Japanese) also appear very early in the morning on /m/. This is suspected to be from actual Japanese posters, as early morning in America is evening in Japan.
The symbols used in Japanese, Chinese, and Korean (and sometimes also other languages that use different character sets from English) are known as "Moon Runes".
While Moonspeak is the universal term for a different language, Spanish has been dubbed Taco Speak.
A reference to the term appears in the videogame Disgaea 2, the description of the rune staff item claims "Some crazy moon language is carved in it"