Lua is a simple lightweight highly portable dynamically typed embeddable and extendable multi-paradigm scripting language. The "vanilla" (some would say official) implementation of it is made purely on ANSI C and has an awesome (simple yet powerful) C API that you can use to both embed Lua on your app or extend the behavior of the language itself. It is developed at the Informatics Department of the Pontifical Catholic University of Rio de Janeiro (PUC-Rio).
Thought it was not primarily designed for that, Lua found a big niche in game scripting, with big names such as "Grim Fandango" and "World of Warcraft". Nonetheless, because of its speed, simplicity and portability, it is also heavily used in embedded systems (see, for example, eLua project) and graphic computing.
Its philosophy is to be minimalistic, i.e its core libraries are very small with only minimum functionality (quite like C's standard libraries), though through the C API it is very simple to add features that wouldn't be possible through the pure core library, such as sockets, GUIs etc. In fact, Lua is so minimalistic that its main -and only- structured data type are 'tables', that could be described as associative arrays on 'roids.
Lua is procedural in its essence, but also supports multiple paradigms such as functional programming and object orientation.
Though Lua is not the fastest scripting language around (probably javascript's V8 project wins the prize) it is very fast (faster than vanilla Python or Ruby, for instance) and also features a non-official just-in-time implementation called LuaJIT.
In the end, Lua is actually no more than a fun language to play with, which I recommend!.
=)
About tutorials, I'd recommend the article about that on the lua-users wiki.
I hope I helped! =)
PS: I couldn't post all the links because I'm new on stack overflow, but it shouldn't be hard to find everything on Google. Sorry. =(
