The standard command line interpreter for Lua is an example of just such a program. On windows, it is a small executable that is linked to lua52.dll. Its source is, of course, part of the Lua distribution.
Despite being located in the same folder as the sources to the Lua DLL, lua.c only references the public API for Lua, and depends only on the four public header files and the DLL itself.
An even simpler example that embeds a Lua interpreter in a C program is the following, derived from the example shown in the PiL book available online:
#include <stdio.h>
#include <string.h>
#include <lua.h>
#include <lauxlib.h>
#include <lualib.h>
int main (void) {
char buff[256];
int error;
lua_State *L = luaL_newstate(); /* create state */
luaL_openlibs(L); /* open standard libraries */
while (fgets(buff, sizeof(buff), stdin) != NULL) {
error = luaL_loadbuffer(L, buff, strlen(buff), "line") ||
lua_pcall(L, 0, 0, 0);
if (error) {
fprintf(stderr, "%s", lua_tostring(L, -1));
lua_pop(L, 1); /* pop error message from the stack */
}
}
lua_close(L);
return 0;
}
In your existing application, you would need to call luaL_newstate() once and store the returned handle. Along with a call to luaL_openlibs(), you would likely want to also define one or more Lua modules representing your application's scriptable API. And of course, you need to call lua_close() sometime before exiting so that Lua has a chance to clean up its objects and in particular a chance to deal with any objects that the script authors are depending on to get resources released when the application exits.
With that in place, you generally provide a way to load script fragments provided by your user using luaL_loadbuffer() or any of several other functions built on top of lua_load(). Loading a script compiles it and leaves an anonymous function on the top of the stack that when called will execute all top-level statements in the script.
For a lot more discussion of this, see the chapters of Programming in Lua (an older addition is available online) that relate to the C API.