Scripting languages were originally thought of as a control mechanisms for applications written in a hard programming language. The compiled programs could not be modified at runtime, so scripting gave people flexibility.
Most notably, shell script was automating processes in the OS kernel (traditionally, AppleScript on Macs); a role which more and more passed into Perl's hands, and then out of it into Python lately. I've seen Scheme (particularly in its Guile implementation) used to declare raytracing scenes; and recently, Lua is very popular as the programming language to script games - to the point that the only hard-coded thing in many new games is the graphics/physics engine, while the whole game logic is encoded in Lua. In the same way, JavaScript was thought to script the behaviour of a web browser.
The languages emancipated; no-one now thinks about the OS as an application (or thinks about it much at all), and many formerly scripting languages began to be used to write full applications of their own. The name itself became meaningless, and spread to many interpreted languages in use today, regardless of whether they are designed to be interpreted from within another system or not.
However, "scripting languages" is most definitely not synonymous with "interpreted languages" - for instance, BASIC was interpreted for most of its life (i.e. before it lost its acronimicity and became Visual Basic), yet no-one really thinks of it as scripting.
UPDATE: Reading material as usual available at Wikipedia.