To test Lua code outside of WoW, yes you would need a Lua interpreter, but if you're testing your code in WoW itself, you don't. Having said that, if you're learning from scratch via the book, I'd do what it says.
WowLua is a WoW-specific syntax highlighting plugin for your text editor. It highlight elements of the WoW API in addition to normal Lua syntax highlighting.
I have mine currently set like the Lua highlighting on these forums: