Success
and complete
are documented at here. You don't really need to worry about what textStatus is passed to the functions, it's all handled automatically. The success function is called when/if the call succeeds, and complete is called at the very end of everything else.
So, as an example from something I'm working on, I chained together three functions that are triggered when you click a section of the page: a div fades out, and when that's done it runs an ajax call. The success function updates the content of the div (while it's faded out), and then the complete function calls fadeIn on the div. So what it does is it fades out, updates, and fades back in with new content.