Other people have given good examples about scripting languages in general. But one thing that Lua in particular excels at, is being embedded in other programs.

Other people have given good examples about scripting languages in general. But one thing that Lua in particular excels at, is being embedded in other programs. For example, it's a common language for video games to use, both for internal scripting and external modding. For internal use, you give the game designers a higher-level and easier-to-use language with quick feedback (no recompiling your game just to change some logic). For modders, it lets you sandbox their contributions to a specific API, but let them implement arbitrary logic inside of it. This lets users install mods from untrusted sources with confidence that it's not going to crash their computer, delete their data or install a keylogger.