I've just installed songbird: it has a fantastic user interface. Then, I've read it's made with XUL, the core of Mozilla Firefox.
Ok, but I don't understand: Songbird is a desktop interface behind a web interface? And What is XUL? A library?
What can I do with XUL? Desktop application GUI/interface like Web application (HTML+JS+...)?
XUL is a markup language designed to describe user interfaces(in many ways similar to XAML). Like you mentioned, XUL is what Firefox is made of(along with JavaScript).
XUL is used for desktop applications(usage of XUL for web applications is likely to be discontinued in Firefox 4). The fact that Songbird is made of XUL doesn't change the fact that it is a desktop application.
Although XUL may look similar to HTML in some ways(both can use JavaScript), the two are not the same. XUL is mostly for desktop applications. HTML is mostly for web applications.
XUL is not "native code". To run XUL applications, you need a "XUL runner".
With enough programming knowledge, what you "can do with XUL" is create applications. It's that simple.