Here is a blog post Cash wrote as a simple intro to web services.  Also, the docs you linked to include more than just authentication-the entire first section explains how to expose a method to the web services API.  

Thanks for the reply

So, I need to develop functions to expose in server side (elgg), there is no functions that are always existing (without developing plugins,  modifiying elgg) and that I can call simply

have any one an enhenced web services api that I can use by installing plug in (in this case, the app will work only for elgg sites that installed the plugin)

I've worked quite a bit with the API framework and it's definitely useful when you get the hang of it. There isn't anything built in, and thats almost the beauty of it. You get to define the functions and customize the functionality. 

I'm not aware of any plugins, but you'd be better off writing your own. Someone else's API is probably useful to them but not necessarily to you.