How would a person use the Lua scripting language to do things such as this:
Opening an application such as IE and controlling it, accessing the C drive and the curr
I'll add to Doug's correct answer the observation that you might be better off not attempting to control IE if the actual goal is just to fetch web pages at the whim of a script. For that, a library like cURL is a much better choice, and there is a binding to it for Lua known as luacurl.
That binding, along with all of the bindings mentioned by Doug are available as part of the Lua for Windows distribution. Reading between the lines, you are most concerned about the Windows platform, so that distribution should be your starting point.
If you plan to integrate Lua with C# and other .NET languages, then you will want to investigate LuaInterface as well.