Lua is designed specifically to provide a scripting/extension language for projects in C or C++, or projects where interop with C is key. This is where it really shines -- where most Lisps expect to be the primary language, but have FFIs to C for performance and integration reasons, Lua
expects to be used for writing plugins, configuration, and the like. It handles those tasks well, and otherwise stays out of the way.
While it's good as a standalone language, it's an outstanding extension language. It's small, trivially portable, and has excellent interop with C. It's not competition for Lisp in general so much as Emacs Lisp or vimscript, designed for systems that have a core written in C but need a lot of customization layered on top.
Configuration for window managers, scripting enemy behavior for game engines written in brutally optimized C++, that kind of thing. Systems with a "hard and soft layer" architecture (http://c2.com/cgi/wiki?AlternateHardAndSoftLayers). (Incidentally, Tcl was also designed with this use case in mind.)
That said, if you're using Lua as your primary language but are proficient in C, you can add a lot of power to it pretty easily. The emphasis on C interop goes both ways. (Also, the implementation is very intelligently engineered, and is worth study if you're interested in language design or virtual machines.)